(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 120787, 2809] NotebookOptionsPosition[ 104901, 2570] NotebookOutlinePosition[ 109150, 2653] CellTagsIndexPosition[ 109070, 2648] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", "FirstSlide", CellTags-> "SlideShowHeader",ExpressionUUID->"ae861ed8-5fa1-4088-84fd-2651cc7b368a"], Cell["Introdu\[CCedilla]\[ATilde]o \[AGrave] F\[IAcute]sica Computacional I \ (2023.2)", "Title", CellChangeTimes->{ 3.559948400406288*^9, {3.699269571766136*^9, 3.699269573578824*^9}, { 3.7099080105038958`*^9, 3.7099080108133383`*^9}, {3.727189496068931*^9, 3.7271894962748613`*^9}, {3.83932299294512*^9, 3.839323000605739*^9}, { 3.874403983135366*^9, 3.87440398322046*^9}, {3.9067874994558992`*^9, 3.906787500518898*^9}},ExpressionUUID->"92508663-47f1-4deb-9615-\ d2c271e8e453"], Cell["Aula 9: n\[UAcute]meros aleat\[OAcute]rios", "Subtitle", CellChangeTimes->{{3.699269582406044*^9, 3.699269584285553*^9}, { 3.8393232649484053`*^9, 3.839323284264048*^9}, 3.840031268565436*^9, { 3.8407028843722677`*^9, 3.840702924713172*^9}, {3.840714600599773*^9, 3.8407146281949797`*^9}, {3.841157909295815*^9, 3.841157911477744*^9}, { 3.841480259563218*^9, 3.8414802737602463`*^9}, {3.8415945589945517`*^9, 3.841594560546528*^9}, {3.842262188714292*^9, 3.8422622084045753`*^9}, { 3.842288735025601*^9, 3.8422887376233883`*^9}, {3.842288769619337*^9, 3.8422887810036163`*^9}, {3.843294467913928*^9, 3.843294500290168*^9}, { 3.843553405919444*^9, 3.8435534093620787`*^9}, {3.843899024345776*^9, 3.8438990498951845`*^9}, {3.844104755695139*^9, 3.8441047787114773`*^9}, { 3.874403985688244*^9, 3.874403985822639*^9}, {3.9067874966928988`*^9, 3.9067874968308997`*^9}, {3.907069815536929*^9, 3.907069833398957*^9}, 3.9073337848014183`*^9, 3.9075087110630875`*^9},ExpressionUUID->"14f8f51b-58e9-4033-aea1-\ fd611ac95bb9"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ee3ea2a1-973b-4e9a-a704-0d8a3c30d1be"], Cell["N\[UAcute]meros pseudoaleat\[OAcute]rios", "Section", CellChangeTimes->{ 3.699269664116198*^9, 3.70454748091461*^9, {3.704547516075143*^9, 3.7045475190812902`*^9}, {3.706294276733098*^9, 3.706294304716371*^9}, { 3.839323324298633*^9, 3.839323331059084*^9}, 3.840031288455207*^9, { 3.840714636688951*^9, 3.840714639748958*^9}, {3.841057130911109*^9, 3.84105713513932*^9}, {3.841480515514193*^9, 3.84148051572683*^9}, { 3.84226221732572*^9, 3.842262223457004*^9}, {3.843294513408573*^9, 3.843294517052754*^9}, {3.8438990711047573`*^9, 3.843899073171992*^9}, { 3.9070698418869634`*^9, 3.9070698482389603`*^9}},ExpressionUUID->"7da6da63-6378-4f26-b58d-\ d42a7250afa4"], Cell["\<\ Existem diversos fen\[OHat]menos em f\[IAcute]sica cuja descri\[CCedilla]\ \[ATilde]o requer invocar eventos aleat\[OAcute]rios, seja pelo \ car\[AAcute]ter intr\[IAcute]nseco desses fen\[OHat]menos (caso, por exemplo, \ do decaimento radioativo), seja pela nossa ignor\[AHat]ncia a respeito de \ todas as vari\[AAcute]veis do problema (caso dos fen\[OHat]menos \ t\[EAcute]rmicos). Lidar com esses fen\[OHat]menos computacionalmente envolve \ produzir n\[UAcute]meros aleat\[OAcute]rios.\ \>", "Text", CellChangeTimes->{{3.6992696952616243`*^9, 3.6992696957779827`*^9}, { 3.704547416597644*^9, 3.704547418388081*^9}, {3.8393233348613234`*^9, 3.839323366214238*^9}, {3.839323576559081*^9, 3.839323576563138*^9}, 3.840031301280365*^9, {3.840714647849332*^9, 3.840714700814382*^9}, { 3.841480523939232*^9, 3.841480544976879*^9}, {3.841480739096553*^9, 3.841480739636643*^9}, {3.8422622643967133`*^9, 3.8422623532037497`*^9}, { 3.842262439308573*^9, 3.8422624573684273`*^9}, {3.8422649573788548`*^9, 3.8422649619920673`*^9}, {3.842358288105185*^9, 3.8423582908266573`*^9}, { 3.8432945364257298`*^9, 3.8432945825057983`*^9}, {3.843294685610161*^9, 3.843294812070178*^9}, {3.8432948733770227`*^9, 3.843294900533929*^9}, { 3.843294935611968*^9, 3.843294945646529*^9}, {3.843294979599766*^9, 3.843295039675002*^9}, {3.843295250118289*^9, 3.84329528363239*^9}, { 3.8435546465204973`*^9, 3.843554650255834*^9}, {3.843640645839772*^9, 3.843640685473322*^9}, {3.843899079600821*^9, 3.8438991439134645`*^9}, { 3.8438995257773323`*^9, 3.843899525781332*^9}, {3.9070699841063557`*^9, 3.907070147389688*^9}},ExpressionUUID->"3b730a40-c305-44bd-95b8-\ df5440115120"] }, Open ]], Cell[TextData[{ "Mais apropriadamente, uma vez que nossos computadores atuais s\[ATilde]o m\ \[AAcute]quinas determin\[IAcute]sticas, trabalhamos com n\[UAcute]meros \ pseudoaleat\[OAcute]rios, ou seja, n\[UAcute]meros que \ \[OpenCurlyDoubleQuote]parecem\[CloseCurlyDoubleQuote] aleat\[OAcute]rios, ao \ menos segundo uma s\[EAcute]rie de testes estat\[IAcute]sticos. Para maiores \ detalhes sobre os algoritmos computacionais que geram esses n\[UAcute]meros, \ consultem, por exemplo, o cap\[IAcute]tulo 10 de ", StyleBox["Computational Physics", FontSlant->"Italic"], ", de Mark Newman. No restante da aula, por simplicidade, vamos nos referir \ a n\[UAcute]meros pseudoaleat\[OAcute]rios apenas como \ \[OpenCurlyDoubleQuote]n\[UAcute]meros aleat\[OAcute]rios\ \[CloseCurlyDoubleQuote]." }], "Text", CellChangeTimes->{{3.8439000487413206`*^9, 3.8439000610855465`*^9}, { 3.843900144204976*^9, 3.8439001839881444`*^9}, {3.843900235028965*^9, 3.8439002427850943`*^9}, {3.843900287048004*^9, 3.843900303636487*^9}, { 3.843900499106572*^9, 3.8439006240234003`*^9}, {3.874405544172724*^9, 3.8744055443623037`*^9}, {3.906787569520356*^9, 3.9067875712947445`*^9}, { 3.907070149484683*^9, 3.9070702408182898`*^9}, {3.9070702745403304`*^9, 3.9070702746753087`*^9}, {3.907070601702413*^9, 3.9070706905726166`*^9}, 3.9070850582529535`*^9, {3.9075031886496606`*^9, 3.9075032023625927`*^9}},ExpressionUUID->"08c06edf-786a-4810-91b9-\ 62b74aaaa484"], Cell[TextData[{ "Como de costume, vamos apresentar aqui algumas fun\[CCedilla]\[OTilde]es do \ Mathematica que podem ser utilizadas para produzir n\[UAcute]meros aleat\ \[OAcute]rios. Uma lista completa pode ser encontrada ", ButtonBox["aqui", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/tutorial/\ RandomNumberGeneration.html"], None}, ButtonNote-> "https://reference.wolfram.com/language/tutorial/RandomNumberGeneration.\ html"], ". " }], "Text", CellChangeTimes->{{3.8439000487413206`*^9, 3.8439000610855465`*^9}, { 3.843900144204976*^9, 3.8439001839881444`*^9}, {3.843900235028965*^9, 3.8439002427850943`*^9}, {3.843900287048004*^9, 3.843900303636487*^9}, { 3.843900499106572*^9, 3.8439006240234003`*^9}, {3.874405544172724*^9, 3.8744055443623037`*^9}, {3.906787569520356*^9, 3.9067875712947445`*^9}, { 3.907070149484683*^9, 3.9070702408182898`*^9}, {3.9070702745403304`*^9, 3.9070702746753087`*^9}, {3.907070601702413*^9, 3.907070757891805*^9}, { 3.907070817119876*^9, 3.9070708283650527`*^9}, {3.9070849609935956`*^9, 3.9070850556951737`*^9}, {3.907349536540259*^9, 3.9073495378419285`*^9}},ExpressionUUID->"761ae640-7dce-4f49-bf8a-\ 98168384f43e"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"0d6bbe14-c226-415e-b1fc-fcf294444606"], Cell[TextData[{ "Se quisermos produzir n\[UAcute]meros inteiros aleat\[OAcute]rios dentro de \ um certo intervalo, a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["RandomInteger", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/RandomInteger.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/RandomInteger.html"], FontFamily->"Source Code Pro"], " \[EAcute] \[UAcute]til. O exemplo abaixo sorteia um inteiro entre ", Cell[BoxData[ FormBox[ RowBox[{"-", "2"}], TraditionalForm]],ExpressionUUID-> "c2066fb9-c377-4b85-90ff-32df7652efda"], " e ", Cell[BoxData[ FormBox["3", TraditionalForm]],ExpressionUUID-> "d9ab24dd-fbd6-429c-90ec-76c0a7fe6a2b"], ":" }], "Text", CellChangeTimes->{{3.9070852814362717`*^9, 3.907085432137502*^9}, { 3.9070855539416294`*^9, 3.907085567224923*^9}},ExpressionUUID->"318c9b45-c5cd-47f9-9cd5-\ 8c3659055849"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RandomInteger", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "3"}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.9070855990689025`*^9, 3.907085652393224*^9}},ExpressionUUID->"007806eb-b54f-4c8b-ab6f-\ 23f2a4cdfa5e"], Cell[TextData[{ "A fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["RandomReal", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/RandomReal.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/RandomReal.html"], FontFamily->"Source Code Pro"], " gera n\[UAcute]meros reais aleat\[OAcute]rios uniformemente distribu\ \[IAcute]dos dentro de um intervalo especificado. O exemplo abaixo sorteia um \ n\[UAcute]mero real entre ", Cell[BoxData[ FormBox["1", TraditionalForm]],ExpressionUUID-> "674c03fa-b235-4504-b06e-90797b472fab"], " e ", Cell[BoxData[ FormBox["2", TraditionalForm]],ExpressionUUID-> "a2e5e744-bbbe-4175-b6fe-1c49ee0856eb"], ":" }], "Text", CellChangeTimes->{ 3.9073339135094914`*^9},ExpressionUUID->"de6707e6-bf1e-47a4-8bf5-\ 9329b7d82fdd"], Cell[BoxData[ RowBox[{ RowBox[{"RandomReal", "[", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], "]"}], " "}]], "Input",ExpressionUUID->\ "8ed81be0-683d-4e9b-a1a4-a691cdf8b293"], Cell["\<\ Note que a cada execu\[CCedilla]\[ATilde]o da instru\[CCedilla]\[ATilde]o o n\ \[UAcute]mero produzido \[EAcute] diferente.\ \>", "ItemParagraph", CellChangeTimes->{{3.8439021928333054`*^9, 3.843902248839401*^9}, { 3.9067876442950673`*^9, 3.906787657240967*^9}, {3.9070852616522503`*^9, 3.907085276304784*^9}, {3.9073339902318707`*^9, 3.907333990245868*^9}},ExpressionUUID->"4f8ec608-0bd4-4039-b2a6-\ 7659f7e46b8f"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"17774e00-e7a4-4e88-b623-1d0211c771fc"], Cell[TextData[{ "Tanto ", StyleBox["RandomReal", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0., 0., 1.]], " quanto ", StyleBox["RandomInteger", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0, 0, 1]], " podem ser invocadas para sortear listas de n\[UAcute]meros \ aleat\[OAcute]rios, pela adi\[CCedilla]\[ATilde]o de um segundo argumento." }], "Text", CellChangeTimes->{{3.84329829620815*^9, 3.843298318549615*^9}, { 3.843298362594396*^9, 3.84329840933783*^9}, {3.843298440613007*^9, 3.843298468809654*^9}, {3.8439022934095707`*^9, 3.8439023384710937`*^9}, { 3.8439122809009743`*^9, 3.843912318328432*^9}, {3.8439123575687723`*^9, 3.843912417385487*^9}, {3.8744094652467527`*^9, 3.874409485641863*^9}, { 3.8744095166927233`*^9, 3.874409578961897*^9}, {3.90708600148013*^9, 3.907086048526485*^9}, {3.907334105460822*^9, 3.907334127313871*^9}, { 3.9073494692121925`*^9, 3.9073494726947193`*^9}, {3.907503215338718*^9, 3.907503253046624*^9}},ExpressionUUID->"0748b206-6537-42a5-be25-\ 2fa6fe5f3073"], Cell["\<\ O exemplo a seguir gera um vetor com 10 n\[UAcute]meros reais \ aleat\[OAcute]rios entre 0 e 1:\ \>", "Text", CellChangeTimes->{{3.907086088225716*^9, 3.907086108406687*^9}, { 3.90708614394442*^9, 3.907086147678733*^9}, {3.9070862096146727`*^9, 3.907086215827854*^9}},ExpressionUUID->"26bc83b2-746c-41d5-965a-\ 58ef74c0b797"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"RandomReal", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", "10"}], "]"}], "//", "MatrixForm", " "}]], "Input", CellChangeTimes->{{3.9070861516141863`*^9, 3.907086162329605*^9}, 3.9070863359291873`*^9, {3.907086366340495*^9, 3.907086376618907*^9}},ExpressionUUID->"b523f117-86bd-4eb8-8446-\ ed225b10fe49"], Cell["\<\ J\[AAcute] este pr\[OAcute]ximo exemplo gera uma matriz de 3 linhas e 4 \ colunas contendo inteiros 0 ou 1 sorteados aleatoriamente:\ \>", "Text", CellChangeTimes->{{3.907086183825178*^9, 3.907086222906762*^9}, { 3.9070862970506644`*^9, 3.907086316807873*^9}},ExpressionUUID->"1df7ef2f-e376-475a-8bda-\ 55dd6fce4608"], Cell[BoxData[ RowBox[{ RowBox[{"RandomInteger", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "4"}], "}"}]}], "]"}], "//", "MatrixForm", " "}]], "Input", CellChangeTimes->{{3.9070863244936976`*^9, 3.907086354189891*^9}, 3.907086409126871*^9},ExpressionUUID->"c564b902-2375-4e5d-8554-\ 453d7dcdd052"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"916aafb6-db56-46e9-abc4-ac7b00e07c2a"], Cell["Distribui\[CCedilla]\[OTilde]es de probabilidade", "Section", CellChangeTimes->{{3.907347295910825*^9, 3.9073473121145887`*^9}, 3.9073485797467546`*^9, 3.9073488415241547`*^9},ExpressionUUID->"372f1222-0387-4b0f-8b00-\ cb08016cff5b"], Cell[TextData[{ "Os n\[UAcute]meros aleat\[OAcute]rios \[OpenCurlyDoubleQuote]uniformemente \ distribu\[IAcute]dos\[CloseCurlyDoubleQuote] gerados pela fun\[CCedilla]\ \[ATilde]o ", StyleBox["RandomReal", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0., 0., 1.]], " trazem implicitamente a no\[CCedilla]\[ATilde]o de distribui\[CCedilla]\ \[ATilde]o de probabilidade. Para entend\[EHat]-la, considere o racioc\ \[IAcute]nio a seguir. " }], "Text", CellChangeTimes->{{3.907334001998883*^9, 3.907334080263447*^9}, { 3.9073341854186068`*^9, 3.9073343174881015`*^9}, {3.9073343926608725`*^9, 3.9073344455361805`*^9}, {3.9073345041911535`*^9, 3.907334526154422*^9}, { 3.9073359410448713`*^9, 3.9073359786058693`*^9}, {3.9073494464128537`*^9, 3.9073494464158535`*^9}, {3.9075033264431953`*^9, 3.9075033289587574`*^9}},ExpressionUUID->"58af660d-ae9b-48b9-8fb5-\ 9b5587513213"], Cell["\<\ Nossa no\[CCedilla]\[ATilde]o intuitiva de probabilidades tem muito a ver com \ contagem. (Essa n\[ATilde]o \[EAcute] a \[UAcute]nica associa\[CCedilla]\ \[ATilde]o poss\[IAcute]vel, mas n\[ATilde]o vamos discutir no\[CCedilla]\ \[OTilde]es mais subjetivas de probabilidades aqui.) Quando dizemos que a \ probabilidade de obter cara no lan\[CCedilla]amento de uma moeda \[EAcute] \ 1/2, h\[AAcute] uma implica\[CCedilla]\[ATilde]o de que, ao \ lan\[CCedilla]armos 100 vezes seguidas uma moeda, esperamos que, em \ m\[EAcute]dia, 50 vezes a moeda caia com a cara para cima. Quando dizemos que \ a probabilidade de obter a face 3 no lan\[CCedilla]amento de um dado de 6 \ faces \[EAcute] 1/6, h\[AAcute] uma implica\[CCedilla]\[ATilde]o de que, ao \ lan\[CCedilla]armos 120 vezes seguidas o dado, esperamos que, em \ m\[EAcute]dia, 20 vezes o dado caia com a face 3 para cima. Mas quando o \ resultado de um experimento envolvendo a sorte pode variar continuamente, de \ modo que o n\[UAcute]mero de poss\[IAcute]veis resultados torna-se infinito, \ a probabilidade de obter exatamente um dado resultado \[EAcute] nula! S\ \[OAcute] faz sentido perguntar sobre a probabilidade de obter um resultado \ que esteja contido em um certo intervalo finito de valores.\ \>", "Text", CellChangeTimes->{{3.9073359608858776`*^9, 3.9073359673278704`*^9}, { 3.907336096487929*^9, 3.907336129744634*^9}, {3.9073462280289125`*^9, 3.907346229213849*^9}, {3.907346260114213*^9, 3.9073462602738094`*^9}, { 3.9077748090724945`*^9, 3.9077748286131344`*^9}},ExpressionUUID->"0e6b480f-fb80-49e5-a8cc-\ 7cea305557e8"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"3d840658-e7d8-403c-867c-a584b277195f"], Cell[TextData[{ "Uma ferramenta matem\[AAcute]tica para visualizar contagens em intervalos s\ \[ATilde]o os histogramas. Vamos produzir um histograma de uma lista de ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["n", "TI"], TraditionalForm], "errors" -> {}, "input" -> "n", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "1e0e4ef6-b9d9-4dc4-a7ef-5e2ba1aa391c"], " n\[UAcute]meros sorteados pela fun\[CCedilla]\[ATilde]o ", StyleBox["RandomReal", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0., 0., 1.]], " no intervalo entre 0 e 1, ou seja, vamos dividir o intervalo em \ \[OpenCurlyDoubleQuote]caixas\[CloseCurlyDoubleQuote] de uma certa largura ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalDelta]", StyleBox["x", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "\\Delta x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "42fae028-bdcb-4a90-8138-d62a39874b40"], " e contar quantos dos n\[UAcute]meros sorteados est\[ATilde]o dentro de \ cada caixa. No Mathematica, isso pode ser feito com a ajuda da fun\[CCedilla]\ \[ATilde]o ", StyleBox[ButtonBox["Histogram", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Histogram.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Histogram.html"], FontFamily->"Source Code Pro"], "." }], "Text", CellChangeTimes->{{3.90733599430587*^9, 3.907336045129916*^9}, { 3.9073361382787075`*^9, 3.9073361397515297`*^9}, {3.907345339157263*^9, 3.90734534823249*^9}, {3.907349404583332*^9, 3.9073494246853857`*^9}, { 3.907486908715722*^9, 3.9074869204747715`*^9}, {3.907503363715355*^9, 3.907503383126749*^9}, {3.907774740411269*^9, 3.90777475350962*^9}},ExpressionUUID->"cfb54ed5-d635-4f66-a835-\ 4544089993db"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"n", "=", "100000"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{"Quantos", " ", "n\[UAcute]meros", " ", "vamos", " ", "sortear"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]x", " ", "=", " ", "0.1"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{"Largura", " ", "da", " ", "caixa"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"lista", "=", RowBox[{"RandomReal", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", "n"}], "]"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{"Sorteamos", " ", "os", " ", "n", " ", "n\[UAcute]meros"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"optam", "=", RowBox[{"{", RowBox[{ RowBox[{"ImageSize", "->", "Large"}], ",", RowBox[{"LabelStyle", "->", "Large"}], ",", RowBox[{"Axes", "->", "False"}], ",", RowBox[{"Frame", "->", "True"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Histogram", "[", RowBox[{"lista", ",", RowBox[{"{", "\[CapitalDelta]x", "}"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "A", " ", "sintaxe", " ", "exige", " ", "chaves", " ", "envolvendo", " ", "\[CapitalDelta]x"}], " ", "*)"}], "\[IndentingNewLine]", "optam", ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ StyleBox[\"x\", \"TI\"], TraditionalForm], \"errors\" -> {}, \"input\" -> \"x\ \", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\"", ",", "\"\\""}], "}"}]}]}], "]"}], " "}]}], "Input", CellChangeTimes->{{3.907334320931423*^9, 3.907334381721596*^9}, { 3.9073344551545563`*^9, 3.9073344875964346`*^9}, {3.907334534760723*^9, 3.9073345457561035`*^9}, {3.907334724687269*^9, 3.9073347614117527`*^9}, { 3.907334809862954*^9, 3.907334966877326*^9}, 3.9073362070726423`*^9, { 3.9073451872208*^9, 3.9073452825709763`*^9}, {3.907345367094121*^9, 3.9073454002219257`*^9}, 3.907507608278023*^9},ExpressionUUID->"58f62f35-7b7a-4c45-ac6b-\ b396aa595c37"], Cell[TextData[{ "Note que h\[AAcute] aproximadamente ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["n", "TI"], "/", "10"}], TraditionalForm], "errors" -> {}, "input" -> "n/10", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "25b72872-02ca-48de-bb86-b54e76a2ddb9"], " pontos dentro de cada uma das ", Cell[BoxData[ FormBox[ TemplateBox[<| "boxes" -> FormBox["10", TraditionalForm], "errors" -> {}, "input" -> "10", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "f98ea321-a9ed-4c9c-8c39-21afb8973d64"], " caixa. Se agora dividirmos por 2 a largura da caixa, a contagem aproximada \ em cada uma das ", Cell[BoxData[ FormBox[ TemplateBox[<| "boxes" -> FormBox["20", TraditionalForm], "errors" -> {}, "input" -> "20", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "f974468f-c4d3-48a4-a82e-20bbac7cf637"], " caixas passa a ser ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["n", "TI"], "/", "20"}], TraditionalForm], "errors" -> {}, "input" -> "n/20", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "8c8a9cfb-3ff5-4ee1-9427-98884c988bc4"], ":" }], "Text", CellChangeTimes->{{3.907334568876896*^9, 3.9073346004348335`*^9}, { 3.9073346518620925`*^9, 3.907334720899437*^9}, {3.907334974262888*^9, 3.9073350316436343`*^9}, {3.907345420568514*^9, 3.9073454240165157`*^9}, { 3.907774774606906*^9, 3.9077747922806406`*^9}},ExpressionUUID->"0ad34ebf-4a00-428a-ab8a-\ 2009945e14cf"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Histogram", "[", RowBox[{"lista", ",", RowBox[{"{", RowBox[{"\[CapitalDelta]x", "/", "2"}], "}"}], ",", "\[IndentingNewLine]", "optam", ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ StyleBox[\"x\", \"TI\"], TraditionalForm], \"errors\" -> {}, \"input\" -> \"x\ \", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\"", ",", "\"\\""}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.9073350444625463`*^9, 3.9073350446221967`*^9}, 3.907345431677642*^9, 3.9075077371225786`*^9},ExpressionUUID->"97f18835-2c87-4985-ab1b-\ 23c2bacf2f05"], Cell["\<\ Que a contagem diminua pela metade quando diminu\[IAcute]mos pela metade a \ largura da caixa \[EAcute] razo\[AAcute]vel, e indica, na verdade, que a \ contagem deve ir a zero quando a largura da caixa for a zero. Afinal, o n\ \[UAcute]mero de seres humanos vivos com altura entre 1,70 m e 1,71 m deve \ ser de muitos milh\[OTilde]es; mas qual \[EAcute] o n\[UAcute]mero de pessoas \ com altura entre 1,70000000 m e 1,70000001 m? \ \>", "Text", CellChangeTimes->{{3.9073350854905305`*^9, 3.9073352431414585`*^9}, { 3.907335277686513*^9, 3.9073354455067234`*^9}, 3.9073354838025217`*^9, { 3.9073454410536313`*^9, 3.9073454431624804`*^9}},ExpressionUUID->"aea8eed7-0764-47a8-be6c-\ 05ab29cb5605"] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"2184bb19-e2a9-4543-bf3c-eceaa64ba3b6"], Cell[TextData[{ "Se quisermos que o histograma mostre n\[ATilde]o a contagem de \ n\[UAcute]meros em cada caixa, mas a ", StyleBox["densidade de probabilidade", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], "(", StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "P(x)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "28447ff7-7316-45d3-ab41-8158ed3e2c98"], " de encontrarmos um n\[UAcute]mero da lista dentro da caixa que se inicia \ no valor ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "7d765ee9-cdad-4860-bcb3-f85917410dc0"], ", podemos agregar a op\[CCedilla]\[ATilde]o \[OpenCurlyDoubleQuote]", StyleBox["PDF", FontFamily->"Source Code Pro"], "\[CloseCurlyDoubleQuote] (de \ \[OpenCurlyDoubleQuote]fun\[CCedilla]\[ATilde]o densidade de probabilidade\ \[CloseCurlyDoubleQuote]) \[AGrave] fun\[CCedilla]\[ATilde]o ", StyleBox["Histogram", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0, 0, 1]], ":" }], "Text", CellChangeTimes->{{3.9073354968455515`*^9, 3.9073359158898907`*^9}, { 3.907345492303797*^9, 3.9073455096564817`*^9}, {3.9073456300162477`*^9, 3.9073457040915766`*^9}, {3.907348468119387*^9, 3.907348473266051*^9}, { 3.9073485857564335`*^9, 3.907348590973727*^9}, {3.9073493711269836`*^9, 3.907349371131981*^9}, {3.907503408161508*^9, 3.9075034515633945`*^9}},ExpressionUUID->"1c0047f6-2cd0-4428-b0ec-\ adb3d7dd2d6b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Histogram", "[", RowBox[{"lista", ",", RowBox[{"{", "\[CapitalDelta]x", "}"}], ",", "\"\\"", ",", " ", RowBox[{"(*", " ", RowBox[{ "Tamanho", " ", "da", " ", "caixa", " ", "\[EAcute]", " ", "0.1"}], " ", "*)"}], "\[IndentingNewLine]", "optam", ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ StyleBox[\"x\", \"TI\"], TraditionalForm], \"errors\" -> {}, \"input\" -> \"x\ \", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\"", ",", "\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ RowBox[{ StyleBox[\"P\", \"TI\"], \"(\", StyleBox[\"x\", \"TI\"], \")\"}], TraditionalForm], \"errors\" -> {}, \"input\ \" -> \"P(x)\", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.907336173290521*^9, 3.907336181726529*^9}, { 3.907336351120108*^9, 3.907336369518359*^9}, {3.9077757566400948`*^9, 3.907775765244049*^9}},ExpressionUUID->"e5e6a13c-c31f-4cfd-a234-\ 471e1081e9c3"], Cell[TextData[{ "Note que agora a densidade de probabilidade em cada intervalo \[EAcute] \ aproximadamente ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], RowBox[{"(", StyleBox["x", "TI"], ")"}], "\[LongEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "P(x)=1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "5eec4e7e-33a1-4d15-8a12-ff99dd41905b"], ", e nesse caso n\[ATilde]o se altera se variarmos o tamanho da caixa:" }], "Text", CellChangeTimes->{{3.9073362596816573`*^9, 3.907336341552433*^9}, { 3.9073473611449423`*^9, 3.9073473629632006`*^9}, {3.9073476698453817`*^9, 3.9073476730423737`*^9}},ExpressionUUID->"0bd98c0e-9b2c-4a34-af52-\ 58f867976cb4"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Histogram", "[", RowBox[{"lista", ",", RowBox[{"{", RowBox[{"\[CapitalDelta]x", "/", "2"}], "}"}], ",", "\"\\"", ",", " ", RowBox[{"(*", " ", RowBox[{ "Tamanho", " ", "da", " ", "caixa", " ", "\[EAcute]", " ", "0.05"}], " ", "*)"}], "\[IndentingNewLine]", "optam", ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ StyleBox[\"x\", \"TI\"], TraditionalForm], \"errors\" -> {}, \"input\" -> \"x\ \", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\"", ",", "\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ RowBox[{ StyleBox[\"P\", \"TI\"], \"(\", StyleBox[\"x\", \"TI\"], \")\"}], TraditionalForm], \"errors\" -> {}, \"input\ \" -> \"P(x)\", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\""}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.907336379920107*^9, 3.907336392933548*^9}, 3.9075077535648336`*^9, {3.907775778227314*^9, 3.9077757842074685`*^9}},ExpressionUUID->"cb9a1503-94e1-45cd-acfb-\ a7664c1e283a"], Cell[TextData[{ "A densidade de probabilidade ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], "(", StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "P(x)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "ba1c2865-6d54-4fc9-a047-5d296264c948"], " ", StyleBox["n\[ATilde]o", FontWeight->"Bold"], " corresponde diretamente \[AGrave] probabilidade de encontrarmos um \ elemento da lista na caixa que se inicia no valor ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "22025c74-843d-42e6-8a7f-0d1870a44142"], ", que \[EAcute] dada de fato por ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], RowBox[{"(", StyleBox["x", "TI"], ")"}], "\[CapitalDelta]", StyleBox["x", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "P(x)\\Delta x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "1acba47c-9808-4037-8421-de8df0e90fd6"], ", quantidade definida pela fra\[CCedilla]\[ATilde]o de elementos da lista \ (no limite em que seu tamanho tende a infinito) que pertencem \[AGrave] \ caixa. " }], "Text", CellChangeTimes->{{3.907345574283184*^9, 3.9073456203284883`*^9}, { 3.907345723495386*^9, 3.9073460695915236`*^9}, {3.9073461083987494`*^9, 3.907346172573481*^9}, 3.9074939753220224`*^9, {3.9075034634826355`*^9, 3.9075034916294403`*^9}, {3.9077748869378223`*^9, 3.907775079934682*^9}, { 3.907775243399132*^9, 3.9077753361332035`*^9}},ExpressionUUID->"24c65018-5f12-43b9-a4e4-\ 69f060b3c829"], Cell[TextData[{ "A probabilidade de encontrarmos um elemento da lista em uma de duas caixas \ cont\[IAcute]guas que se iniciam em ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox[ StyleBox["x", "TI"], "1"], TraditionalForm], "errors" -> {}, "input" -> "x_1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "3bab5989-eca5-4107-9e08-c7cb486c9c17"], " e ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ StyleBox["x", "TI"], "2"], "\[LongEqual]", SubscriptBox[ StyleBox["x", "TI"], "1"], "+", "\[CapitalDelta]", StyleBox["x", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "x_2=x_1+\\Delta x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "26644a54-6b6f-4ea3-8ac8-4ff51dbadf35"], ", por sua vez, \[EAcute] estimada pela soma das fra\[CCedilla]\[OTilde]es \ de elementos em cada caixa, o que corresponde a ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["x", "TI"], "1"], ")"}], "\[CapitalDelta]", StyleBox["x", "TI"], "+", StyleBox["P", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["x", "TI"], "2"], ")"}], "\[CapitalDelta]", StyleBox["x", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "P(x_1)\\Delta x+P(x_2)\\Delta x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "7b989bd4-a892-4087-b72a-906f483159b5"], ", e analogamente para uma quantidade maior de caixas cont\[IAcute]guas." }], "Text", CellChangeTimes->{{3.907345574283184*^9, 3.9073456203284883`*^9}, { 3.907345723495386*^9, 3.9073460695915236`*^9}, {3.9073461083987494`*^9, 3.907346172573481*^9}, 3.9074939753220224`*^9, {3.9075034634826355`*^9, 3.9075034916294403`*^9}, {3.9077748869378223`*^9, 3.9077750567926903`*^9}, {3.907775091341247*^9, 3.9077752100024633`*^9}, { 3.907775349050893*^9, 3.907775361043524*^9}},ExpressionUUID->"e5a7aabc-8325-4109-97f8-\ 2b7969e6209b"], Cell[TextData[{ "A soma de ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], RowBox[{"(", StyleBox["x", "TI"], ")"}], "\[CapitalDelta]", StyleBox["x", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "P(x)\\Delta x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "1a507cd8-bcc3-4fd8-afa9-68002f9e72b7"], " sobre todas as caixas, que corresponde \[AGrave] probabilidade de \ encontrar um n\[UAcute]mero da lista em alguma das caixas, \[EAcute] igual a \ 1. No limite em que ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalDelta]", StyleBox["x", "TI"], "\[Rule]", "0"}], TraditionalForm], "errors" -> {}, "input" -> "\\Delta x\\rightarrow 0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "42b1ff08-3af0-4753-b2b3-48afdf25af22"], " e o n\[UAcute]mero de elementos da lista tende ao infinito, essa ", StyleBox["condi\[CCedilla]\[ATilde]o de normaliza\[CCedilla]\[ATilde]o", FontWeight->"Bold"], " se torna" }], "Text", CellChangeTimes->{{3.907345574283184*^9, 3.9073456203284883`*^9}, { 3.907345723495386*^9, 3.9073460695915236`*^9}, {3.9073461083987494`*^9, 3.907346172573481*^9}, 3.9074939753220224`*^9, {3.9075034634826355`*^9, 3.9075034916294403`*^9}, {3.9077748869378223`*^9, 3.9077750567926903`*^9}, {3.907775091341247*^9, 3.9077751607917137`*^9}, { 3.9077756339187617`*^9, 3.907775650530418*^9}},ExpressionUUID->"3c1ea690-c33e-474e-a263-\ 645ee531c4bd"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Integral]", StyleBox["P", "TI"], RowBox[{"(", StyleBox["x", "TI"], ")"}], StyleBox["d", "TI"], StyleBox["x", "TI"], "\[LongEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "\\displaystyle \\int P(x)dx = 1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], ","}], TraditionalForm]],ExpressionUUID->"24a384c2-6fc3-4b06-8b99-ea1f288869c8"]], \ "Text", CellChangeTimes->{{3.907345574283184*^9, 3.9073456203284883`*^9}, { 3.907345723495386*^9, 3.9073460695915236`*^9}, {3.9073461083987494`*^9, 3.907346172573481*^9}, 3.9074939775923586`*^9},ExpressionUUID->"3a6a96ea-2acb-4466-954b-\ 1736e0f98d0c"], Cell[TextData[{ "com a integral tomada sobre todo o intervalo de valores permitidos de ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "3813542d-8402-4011-aea2-03cadcfa5ec8"], ", que aqui corresponde a ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"0", "\[LessEqual]", StyleBox["x", "TI"], "\[LessEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "0\\leq x\\leq 1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "274cbc5a-4f23-4256-a3a7-2c76c7b4bb35"], "." }], "Text", CellChangeTimes->{{3.907345574283184*^9, 3.9073456203284883`*^9}, { 3.907345723495386*^9, 3.9073460695915236`*^9}, {3.9073461083987494`*^9, 3.907346172573481*^9}, 3.9077753954696217`*^9},ExpressionUUID->"c3e71855-7766-4564-bfe7-\ 982a5985e65b"], Cell[TextData[{ "Nesse mesmo limite, o racioc\[IAcute]nio sobre as caixas cont\[IAcute]guas \ mostra que a probabilidade de encontrarmos um n\[UAcute]mero da lista com \ valores entre ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["a", "TI"], TraditionalForm], "errors" -> {}, "input" -> "a", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "4e4140c1-f69b-49eb-b9de-9480d62520f3"], " e ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["b", "TI"], TraditionalForm], "errors" -> {}, "input" -> "b", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "8dbb29cf-bd07-497e-b2b1-5f301e4013af"], " \[EAcute] " }], "Text", CellChangeTimes->{{3.907345574283184*^9, 3.9073456203284883`*^9}, { 3.907345723495386*^9, 3.9073460695915236`*^9}, {3.9073461083987494`*^9, 3.907346172573481*^9}, {3.907775396173768*^9, 3.9077754281153393`*^9}, { 3.9077754684890547`*^9, 3.9077754999460573`*^9}, {3.907775584611843*^9, 3.90777558743187*^9}},ExpressionUUID->"c850481b-5cd9-429c-9b6e-\ 9eee7cff7140"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox["a", "TI"], StyleBox["b", "TI"]], StyleBox["P", "TI"], RowBox[{"(", StyleBox["x", "TI"], ")"}], StyleBox["d", "TI"], StyleBox["x", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "\\displaystyle \\int_a^b P(x)dx", "state" -> "Boxes"|>, "TeXAssistantTemplate"], "."}], TraditionalForm]],ExpressionUUID->"7a7356e1-ed04-4b55-a1e2-16eeac832459"]], \ "Text", CellChangeTimes->{{3.907345574283184*^9, 3.9073456203284883`*^9}, { 3.907345723495386*^9, 3.9073460695915236`*^9}, {3.9073461083987494`*^9, 3.907346172573481*^9}, 3.9074939775923586`*^9, {3.907775598543914*^9, 3.907775599228593*^9}},ExpressionUUID->"62fc68c2-ac58-48cc-8323-\ c69fe7b9e8a1"], Cell[TextData[{ "O conjunto de resultados poss\[IAcute]veis de um experimento envolvendo \ vari\[AAcute]veis aleat\[OAcute]rias \[EAcute] chamado de ", StyleBox["espa\[CCedilla]o amostral", FontWeight->"Bold"], ". No caso de experimentos cujo espa\[CCedilla]o amostral seja discreto, a \ distribui\[CCedilla]\[ATilde]o de probabilidades \[EAcute] a fun\[CCedilla]\ \[ATilde]o que retorna a probabilidade de cada poss\[IAcute]vel resultado. J\ \[AAcute] no caso de espa\[CCedilla]os amostrais cont\[IAcute]nuos, \ h\[AAcute] quem chame de distribui\[CCedilla]\[ATilde]o de probabilidades a \ densidade de probabilidade integrada, ", StyleBox["mas aqui vamos quase sempre usar densidade ou distribui\[CCedilla]\ \[ATilde]o de probabilidade de forma indistinta", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.907348084479546*^9, 3.907348144967631*^9}, { 3.9073482134200974`*^9, 3.9073482460713267`*^9}, {3.9073483028073683`*^9, 3.907348339245796*^9}, {3.907348443706117*^9, 3.9073485610151463`*^9}, 3.907348827819018*^9, {3.9073491364680357`*^9, 3.907349149205991*^9}, { 3.907487074237096*^9, 3.90748707889192*^9}, {3.9075035050054226`*^9, 3.9075035379664164`*^9}},ExpressionUUID->"d7f60c65-453d-454f-a021-\ 30918ba149e9"] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"b8cad7d2-e68b-423c-9e00-61e52454936d"], Cell["Distribui\[CCedilla]\[OTilde]es de probabilidades no Mathematica", \ "Section", CellChangeTimes->{{3.907347456116124*^9, 3.907347476896841*^9}, 3.907348568325116*^9},ExpressionUUID->"62e11d5b-18e4-4093-91fe-\ c7d654f80073"], Cell[TextData[{ "O Mathematica tem fun\[CCedilla]\[OTilde]es intr\[IAcute]nsecas para lidar \ com muitas distribui\[CCedilla]\[OTilde]es de probabilidade, como a distribui\ \[CCedilla]\[ATilde]o gaussiana ou normal (", StyleBox[ButtonBox["NormalDistribution", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/NormalDistribution.html"], None}, ButtonNote-> "https://reference.wolfram.com/language/ref/NormalDistribution.html"], FontFamily->"Source Code Pro"], ") e a distribui\[CCedilla]\[ATilde]o uniforme (", StyleBox[ButtonBox["UniformDistribution", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/UniformDistribution.html"]\ , None}, ButtonNote-> "https://reference.wolfram.com/language/ref/UniformDistribution.html"], FontFamily->"Source Code Pro"], "). Essas fun\[CCedilla]\[OTilde]es s\[ATilde]o normalmente usadas em \ combina\[CCedilla]\[ATilde]o com outras, como a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["PDF", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PDF.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PDF.html"], FontFamily->"Source Code Pro"], ", que produz a densidade de probabilidade correspondente, como vamos \ discutir adiante." }], "Text", CellChangeTimes->{{3.907348800471531*^9, 3.9073488234840813`*^9}, { 3.9073488720590973`*^9, 3.907348872685011*^9}, {3.907348936989585*^9, 3.9073489833870263`*^9}, {3.9073490179310503`*^9, 3.9073490439868336`*^9}, {3.907349161482094*^9, 3.907349298161992*^9}, { 3.907349329893916*^9, 3.907349334899193*^9}, 3.9073495754516363`*^9, { 3.907349609244953*^9, 3.907349659361032*^9}, {3.9073497129850416`*^9, 3.9073497129880505`*^9}, {3.907491758766491*^9, 3.907491781615703*^9}, { 3.9075016365473576`*^9, 3.9075016658391614`*^9}, {3.9077922654577923`*^9, 3.907792283874749*^9}},ExpressionUUID->"c9f668d1-4168-469f-8ddc-\ a4c55253873c"], Cell[TextData[{ "A distribui\[CCedilla]\[ATilde]o gaussiana \[EAcute] muito importante em \ matem\[AAcute]tica e nas ci\[EHat]ncias em geral, por ra\.08z\[OTilde]es que \ discutiremos mais tarde. Ela \[EAcute] caracterizada por dois \ par\[AHat]metros: a m\[EAcute]dia ", Cell[BoxData[ FormBox[ TemplateBox[<| "boxes" -> FormBox["\[Mu]", TraditionalForm], "errors" -> {}, "input" -> "\\mu", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "8b99e743-960a-4798-8efa-1f799d8b3c30"], " e o desvio-padr\[ATilde]o ", Cell[BoxData[ FormBox[ TemplateBox[<| "boxes" -> FormBox["\[Sigma]", TraditionalForm], "errors" -> {}, "input" -> "\\sigma", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "4fb0c702-76ad-4bf6-8295-12b8ef2c3241"], ". A densidade de probabilidade ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], "(", StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "P(x)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "900a9ef6-3362-47dd-8fe9-f8b5f154be93"], " correspondente \[EAcute]" }], "Text", CellChangeTimes->{{3.907350025040126*^9, 3.907350071289716*^9}, { 3.907350352837598*^9, 3.907350408747694*^9}, {3.9073504423189173`*^9, 3.907350460863652*^9}, {3.9073504992142096`*^9, 3.9073505171414843`*^9}, { 3.9073506174922523`*^9, 3.9073506191263223`*^9}},ExpressionUUID->"3ea8910b-582c-4377-bd6f-\ c281c0e77b3d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"PDF", "[", RowBox[{ RowBox[{"NormalDistribution", "[", RowBox[{"\[Mu]", ",", "\[Sigma]"}], "]"}], ",", "x"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.9073504639854193`*^9, 3.9073504935705986`*^9}, 3.907507806430468*^9, 3.9077758589534473`*^9},ExpressionUUID->"63f82ab0-de6c-4a6e-8361-\ 6b9cb59214e4"], Cell[TextData[{ "Note que o argumento ", StyleBox["x", FontFamily->"Source Code Pro", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " aparece vinculado \[AGrave] fun\[CCedilla]\[ATilde]o ", StyleBox["PDF", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0, 0, 1]], ", e n\[ATilde]o \[AGrave] fun\[CCedilla]\[ATilde]o ", StyleBox["NormalDistribution", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0, 0, 1]], ". Nesse sentido, essa \[UAcute]ltima fun\[CCedilla]\[ATilde]o (e outras \ semelhantes) tem o papel de uma \[OpenCurlyDoubleQuote]distribui\[CCedilla]\ \[ATilde]o abstrata\[CloseCurlyDoubleQuote], associada pela fun\[CCedilla]\ \[ATilde]o ", StyleBox["PDF", FontFamily->"Source Code Pro", FontWeight->"Normal", FontColor->RGBColor[0, 0, 1]], " a uma vari\[AAcute]vel aleat\[OAcute]ria ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "450ad165-d198-454f-b4a0-42d6417bec61"], "." }], "Text", CellChangeTimes->{{3.9073504639854193`*^9, 3.9073504935705986`*^9}, 3.907507806430468*^9, {3.9077758589534473`*^9, 3.907775887862052*^9}, { 3.907775932350746*^9, 3.907776025617298*^9}, {3.907776116146651*^9, 3.907776141282999*^9}},ExpressionUUID->"89a89ef6-ef0e-4ace-89d4-\ be9a2477f306"], Cell["\<\ Vamos visualizar gr\[AAcute]ficos dessa densidade de probabilidade para dois \ valores diferentes de m\[EAcute]dia e desvio-padr\[ATilde]o:\ \>", "Text", CellChangeTimes->{{3.907350546331279*^9, 3.9073506125925484`*^9}},ExpressionUUID->"41c9c576-c68e-4a8c-a706-\ 68858e84114f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"PDF", "[", RowBox[{ RowBox[{"NormalDistribution", "[", RowBox[{"0", ",", "2"}], "]"}], ",", "x"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"PDF", "[", RowBox[{ RowBox[{"NormalDistribution", "[", RowBox[{"5", ",", "1"}], "]"}], ",", "x"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ StyleBox[\"x\", \"TI\"], TraditionalForm], \"errors\" -> {}, \"input\" -> \"x\ \", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\"", ",", "\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ RowBox[{ StyleBox[\"P\", \"TI\"], \"(\", StyleBox[\"x\", \"TI\"], \")\"}], TraditionalForm], \"errors\" -> {}, \"input\ \" -> \"P(x)\", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\""}], "}"}]}], ",", RowBox[{"PlotLabels", "->", RowBox[{"Placed", "[", RowBox[{ RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ RowBox[{\"\[Mu]\", \"\[LongEqual]\", \"0\", \",\", \"\[Sigma]\", \"\ \[LongEqual]\", \"2\"}], TraditionalForm], \"errors\" -> {}, \"input\" -> \ \"\\\\mu=0,\\\\sigma=2\", \"state\" -> \ \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\"", ",", "\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ RowBox[{\"\[Mu]\", \"\[LongEqual]\", \"5\", \",\", \"\[Sigma]\", \"\ \[LongEqual]\", \"1\"}], TraditionalForm], \"errors\" -> {}, \"input\" -> \ \"\\\\mu=5,\\\\sigma=1\", \"state\" -> \ \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\""}], "}"}], ",", "Above"}], "]"}]}], ",", RowBox[{"Filling", "->", "Axis"}]}], "]"}]], "Input", CellChangeTimes->{{3.907350622724283*^9, 3.907350834184948*^9}, { 3.907350889614416*^9, 3.907350893513638*^9}, {3.9073509807465897`*^9, 3.907351013406334*^9}},ExpressionUUID->"260226cb-209a-435c-a2a7-\ 0f9d9693dfb7"], Cell[TextData[{ "Como se v\[EHat] pelo gr\[AAcute]fico, a m\[EAcute]dia ", Cell[BoxData[ FormBox[ TemplateBox[<| "boxes" -> FormBox["\[Mu]", TraditionalForm], "errors" -> {}, "input" -> "\\mu", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "bb39b993-b15f-427e-8be7-6c07702b9535"], " indica a posi\[CCedilla]\[ATilde]o do m\[AAcute]ximo (sim\[EAcute]trico) \ da distribui\[CCedilla]\[ATilde]o gaussiana ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["P", "TI"], "(", StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "P(x)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "828cd9aa-9d91-4ab6-8e48-bb899ac9a7bc"], " e ", Cell[BoxData[ FormBox[ TemplateBox[<| "boxes" -> FormBox["\[Sigma]", TraditionalForm], "errors" -> {}, "input" -> "\\sigma", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "9f501310-1c9b-46cb-9c1f-f9220c35c02d"], " \[EAcute] uma medida de sua largura. Os nomes \[OpenCurlyDoubleQuote]m\ \[EAcute]dia\[CloseCurlyDoubleQuote] e \[OpenCurlyDoubleQuote]desvio-padr\ \[ATilde]o\[CloseCurlyDoubleQuote] correspondem precisamente \[AGrave] sua \ defini\[CCedilla]\[ATilde]o estat\[IAcute]stica, uma vez que, para a \ distribui\[CCedilla]\[ATilde]o gaussiana, " }], "Text", CellChangeTimes->{{3.907489489300579*^9, 3.907489550898836*^9}, { 3.9074895842288103`*^9, 3.9074896561414223`*^9}, {3.9074897510027275`*^9, 3.9074897554107156`*^9}, {3.9074897884792037`*^9, 3.9074897952881684`*^9}, {3.907489967032246*^9, 3.9074899805391865`*^9}, 3.90749394005511*^9},ExpressionUUID->"797ed65c-020f-4455-b117-\ d03363b4b19d"], Cell[TextData[Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ RowBox[{"<", StyleBox["x", "TI"], ">"}], "\[LongEqual]", SubsuperscriptBox["\[Integral]", RowBox[{"-", "\[Infinity]"}], RowBox[{"+", "\[Infinity]"}]], StyleBox["x", "TI"], StyleBox["P", "TI"], RowBox[{"(", StyleBox["x", "TI"], ")"}], StyleBox["d", "TI"], 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utilizada para \ gerar os n\[UAcute]meros da fun\[CCedilla]\[ATilde]o ", StyleBox["RandomReal", FontFamily->"Source Code Pro", FontColor->RGBColor[0, 0, 1]], ", e pode ser invocada pela sintaxe exemplificada abaixo:" }], "Text", CellChangeTimes->{{3.9073495892725677`*^9, 3.9073495944305873`*^9}, { 3.907501685897994*^9, 3.9075017222995515`*^9}},ExpressionUUID->"f0229977-3958-48ca-a3c1-\ 31585e9be2b4"], Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"PDF", "[", RowBox[{ RowBox[{"UniformDistribution", "[", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], "]"}], ",", "x"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "0.5"}], ",", "1.5"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ StyleBox[\"x\", \"TI\"], TraditionalForm], \"errors\" -> {}, \"input\" -> \"x\ \", \"state\" -> 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3.907490481020938*^9}},ExpressionUUID->"97c3d08c-81dd-4b4e-b3f0-\ a6b29dd364d5"], Cell["\<\ Batizados em refer\[EHat]ncia ao cassino no famoso principado de \ M\[OHat]naco, os m\[EAcute]todos de Monte Carlo constituem essencialmente uma \ ferramenta para calcular numericamente integrais multidimensionais \ complicadas, seguindo algum tipo de amostragem aleat\[OAcute]ria. Aqui vamos \ apresentar apenas alguns exemplos introdut\[OAcute]rios, mas disciplinas avan\ \[CCedilla]adas (como M\[EAcute]todos Computacionais em F\[IAcute]sica) \ abordam aplica\[CCedilla]\[OTilde]es bem mais sofisticadas.\ \>", "Text", CellChangeTimes->{{3.84393082598948*^9, 3.843930827518505*^9}, { 3.843931302776829*^9, 3.843931329214821*^9}, {3.843931448015914*^9, 3.843931455174438*^9}, {3.8439316485591917`*^9, 3.843931757718684*^9}, { 3.844004150101653*^9, 3.84400426326521*^9}, {3.844004298382189*^9, 3.84400429838899*^9}, {3.9067893804037*^9, 3.9067893908198576`*^9}, { 3.907490644277527*^9, 3.9074907872127075`*^9}, {3.907491896618511*^9, 3.9074919099376984`*^9}, {3.907492309500596*^9, 3.9074923208408537`*^9}},ExpressionUUID->"5ef29a2a-0330-4988-8c52-\ 173723bc616f"], Cell[TextData[{ "A ideia b\[AAcute]sica do m\[EAcute]todo pode ser ilustrada com o \ c\[AAcute]lculo da \[AAcute]rea de um c\[IAcute]rculo de raio \ unit\[AAcute]rio, o que equivale a um c\[AAcute]lculo do valor de \[Pi]. A \ equa\[CCedilla]\[ATilde]o de uma circunfer\[EHat]ncia de raio 1 centrada na \ origem \[EAcute] ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SuperscriptBox[ StyleBox["x", "TI"], "2"], "+", SuperscriptBox[ StyleBox["y", "TI"], "2"], "\[LongEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "x^2+y^2=1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "3dbfcf2a-fe6b-478a-b432-a77efd9adfdf"], ", de modo que devemos ter" }], "Text", CellChangeTimes->{{3.9074919201490717`*^9, 3.9074919214862757`*^9}, { 3.907491961262018*^9, 3.9074920675911984`*^9}, {3.9074921150390387`*^9, 3.9074921943095827`*^9}, {3.907492260196019*^9, 3.907492279821692*^9}, { 3.907492335129509*^9, 3.9074923369940634`*^9}, {3.9074925706044035`*^9, 3.9074925780942974`*^9}, {3.9074929046240616`*^9, 3.9074929284561777`*^9}, {3.907493590411628*^9, 3.9074935950882187`*^9}, 3.9074939251104193`*^9},ExpressionUUID->"a16462ec-2187-48a1-b132-\ ef998904a966"], Cell[TextData[Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Pi]", "\[LongEqual]", "4", SubsuperscriptBox["\[Integral]", "0", "1"], SqrtBox[ RowBox[{"1", "-", SuperscriptBox[ StyleBox["x", "TI"], "2"]}]], StyleBox["d", "TI"], StyleBox["x", "TI"], "."}], TraditionalForm], "errors" -> {}, "input" -> "\\displaystyle\\pi=4\\int_0^1\\sqrt{1-x^2}dx.", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID->"0198a347-f00e-4a8a-8f15-bf15bb3e62d4"]], \ "Text", CellChangeTimes->{{3.9074919201490717`*^9, 3.9074919214862757`*^9}, { 3.907491961262018*^9, 3.9074920675911984`*^9}, {3.9074921150390387`*^9, 3.9074921943095827`*^9}, {3.907492260196019*^9, 3.907492279821692*^9}, { 3.907492335129509*^9, 3.9074923369940634`*^9}, {3.9074925706044035`*^9, 3.9074925780942974`*^9}, {3.9074929046240616`*^9, 3.9074929284561777`*^9}, {3.907493590411628*^9, 3.9074935950882187`*^9}, 3.9074939277658243`*^9},ExpressionUUID->"f2d11c6f-971c-40ff-89a4-\ fb8b1d625a54"], Cell[TextData[{ "Um poss\[IAcute]vel c\[AAcute]lculo da integral pelo m\[EAcute]todo de \ Monte Carlo consiste em sortear ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["n", "TI"], TraditionalForm], "errors" -> {}, "input" -> "n", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "339984e2-d268-48e1-bc76-7017b467cbfe"], " pontos no quadrado definido por ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"0", "\[LessEqual]", StyleBox["x", "TI"], "\[LessEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "0\\leq x\\leq 1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "d1035987-352b-4dfd-aa8a-5e8f83de54f4"], " e ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"0", "\[LessEqual]", StyleBox["y", "TI"], "\[LessEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "0\\leq y\\leq 1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "fef1cc57-0938-41e2-b145-650f308cc9e3"], " e determinar a fra\[CCedilla]\[ATilde]o desses pontos sob a curva do \ integrando, que deve ser proporcional \[AGrave] \[AAcute]rea sob o \ gr\[AAcute]fico, conforme ilustrado na figura a seguir." }], "Text", CellChangeTimes->{{3.9074919201490717`*^9, 3.9074919214862757`*^9}, { 3.907491961262018*^9, 3.9074920675911984`*^9}, {3.9074921150390387`*^9, 3.9074921943095827`*^9}, {3.907492260196019*^9, 3.907492279821692*^9}, { 3.907492335129509*^9, 3.9074923369940634`*^9}, {3.9074925706044035`*^9, 3.9074925780942974`*^9}, {3.9074929046240616`*^9, 3.9074929284561777`*^9}, { 3.907493590411628*^9, 3.9074935950882187`*^9}},ExpressionUUID->"c5161276-41fe-4a68-bdb0-\ 5d2bb44230e4"], Cell[BoxData[{ RowBox[{ RowBox[{"n", "=", "1000"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"pontos", "=", RowBox[{"RandomReal", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"n", ",", 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"1"}], "}"}], ",", RowBox[{"PlotStyle", "->", "Red"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"AspectRatio", "->", "1"}]}], "\[IndentingNewLine]", "]"}]}], "Input", CellChangeTimes->{{3.907492340551521*^9, 3.907492549883894*^9}, 3.9075080648745475`*^9},ExpressionUUID->"a8c3a798-507f-47a4-8efa-\ 5ead87b83443"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"0e4ed51b-6e4d-4bf7-b9e4-9aecccabb67e"], Cell[TextData[{ "Em vez de utilizar essa abordagem, vamos fazer uso de uma outra, que \ \[EAcute] baseada na estimativa do valor m\[EAcute]dio do integrando no \ intervalo de integra\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["a", "TI"], "\[LessEqual]", StyleBox["x", "TI"], "\[LessEqual]", StyleBox["b", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "a\\leq x\\leq b", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "07eccfa8-dc0e-42b7-9129-0d1e7339d39b"], "," 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TraditionalForm]],ExpressionUUID->"5335d2ed-b692-4b96-982d-bfa0b8feafea"]], \ "Text", CellChangeTimes->{{3.907492626996236*^9, 3.907492641458103*^9}, { 3.907492701343745*^9, 3.9074927020689173`*^9}, {3.9074930551784263`*^9, 3.9074930748651843`*^9}, {3.907493164808448*^9, 3.9074933602943697`*^9}, { 3.90749345376752*^9, 3.90749345376752*^9}, {3.90749362079399*^9, 3.9074936228116426`*^9}, {3.9074936801457877`*^9, 3.907493689572369*^9}, 3.9074938903857884`*^9},ExpressionUUID->"79c4dbe2-bcc7-425e-b35a-\ bbddb2e246cc"], Cell[TextData[{ "Logo, sendo ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["I", "TI"], TraditionalForm], "errors" -> {}, "input" -> "I", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "d544db21-b78b-4fb0-b09a-45337f558f7b"], " o valor da integral, temos" }], "Text", CellChangeTimes->{{3.907492626996236*^9, 3.907492641458103*^9}, { 3.907492701343745*^9, 3.9074927020689173`*^9}, {3.9074930551784263`*^9, 3.9074930748651843`*^9}, {3.907493164808448*^9, 3.9074933602943697`*^9}, { 3.90749345376752*^9, 3.90749345376752*^9}, {3.90749362079399*^9, 3.9074936228116426`*^9}, {3.9074936801457877`*^9, 3.907493689572369*^9}, 3.9074938950561953`*^9},ExpressionUUID->"370a694f-2e85-4ff7-b66f-\ 21785dc37c14"], Cell[TextData[Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["I", "TI"], "\[LongEqual]", RowBox[{"(", RowBox[{ StyleBox["b", "TI"], "-", StyleBox["a", "TI"]}], ")"}], RowBox[{"<", StyleBox["f", "TI"], ">"}], "."}], TraditionalForm], "errors" -> {}, "input" -> "\\displaystyle I=(b-a)\\left.", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID->"55f433dc-3688-492e-8aca-efde87b43e6e"]], \ "Text", CellChangeTimes->{{3.907492626996236*^9, 3.907492641458103*^9}, { 3.907492701343745*^9, 3.9074927020689173`*^9}, {3.9074930551784263`*^9, 3.9074930748651843`*^9}, {3.907493164808448*^9, 3.9074933602943697`*^9}, { 3.90749345376752*^9, 3.90749345376752*^9}, {3.90749362079399*^9, 3.9074936228116426`*^9}, {3.9074936801457877`*^9, 3.907493689572369*^9}, 3.907493897750831*^9},ExpressionUUID->"a69d362f-3158-4944-ba9f-\ fee80ebe706b"], Cell[TextData[{ "Ocorre que o valor m\[EAcute]dio da fun\[CCedilla]\[ATilde]o pode ser \ estimado sorteando ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["n", "TI"], TraditionalForm], "errors" -> {}, "input" -> "n", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "02c3255f-c0e5-4980-a99a-d187d1fccb52"], " valores ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox[ StyleBox["x", "TI"], StyleBox["i", "TI"]], TraditionalForm], "errors" -> {}, "input" -> "x_i", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "08be7bc0-1aa8-404e-a036-7313cc179edd"], " de ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "57abfbb8-35fc-4669-be13-686aa2916d88"], " uniformemente no intervalo de integra\[CCedilla]\[ATilde]o:" }], "Text", CellChangeTimes->{{3.907492626996236*^9, 3.907492641458103*^9}, { 3.907492701343745*^9, 3.9074927020689173`*^9}, {3.9074930551784263`*^9, 3.9074930748651843`*^9}, {3.907493164808448*^9, 3.9074933602943697`*^9}, { 3.90749345376752*^9, 3.90749345376752*^9}, {3.90749362079399*^9, 3.9074936228116426`*^9}, {3.9074936801457877`*^9, 3.907493689572369*^9}, 3.9074939020085154`*^9, {3.907494380117032*^9, 3.907494388973169*^9}, 3.9074960713240495`*^9},ExpressionUUID->"fd31d79f-e4cd-418e-87d5-\ d0c125ae037c"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ RowBox[{"<", StyleBox["f", "TI"], ">"}], "\[TildeEqual]", FractionBox["1", StyleBox["n", "TI"]], UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["i", "TI"], "\[LongEqual]", "1"}], StyleBox["n", "TI"], LimitsPositioning -> False], StyleBox["f", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["x", "TI"], StyleBox["i", "TI"]], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\displaystyle \\left \\simeq \\frac{1}{n}\\sum_{i=1}^n \ f(x_i)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], "."}], TraditionalForm]],ExpressionUUID->"67f553e4-81ae-4f52-91ba-f5700e2f8321"]], \ "Text", CellChangeTimes->{{3.907492626996236*^9, 3.907492641458103*^9}, { 3.907492701343745*^9, 3.9074927020689173`*^9}, {3.9074930551784263`*^9, 3.9074930748651843`*^9}, {3.907493164808448*^9, 3.9074933602943697`*^9}, { 3.90749345376752*^9, 3.90749345376752*^9}, {3.90749362079399*^9, 3.9074936228116426`*^9}, {3.9074936801457877`*^9, 3.907493689572369*^9}, 3.9074961090091066`*^9},ExpressionUUID->"f138b205-8633-4dff-ac03-\ 0ee73433f9b7"], Cell[TextData[{ "Vamos ent\[ATilde]o utilizar esse m\[EAcute]todo para estimar o valor de ", Cell[BoxData[ FormBox[ TemplateBox[<| "boxes" -> FormBox["\[Pi]", TraditionalForm], "errors" -> {}, "input" -> "\\pi", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "dde87a69-1d90-4689-ba9c-af46a66e2e1d"], ", fazendo uso de opera\[CCedilla]\[OTilde]es com listas e da fun\[CCedilla]\ \[ATilde]o ", StyleBox[ButtonBox["Mean", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Mean.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Mean.html"], FontFamily->"Source Code Pro"], ", que fornece a m\[EAcute]dia dos elementos de uma lista:" }], "Text", CellChangeTimes->{{3.907492626996236*^9, 3.907492641458103*^9}, { 3.907492701343745*^9, 3.9074927020689173`*^9}, {3.9074930551784263`*^9, 3.9074930748651843`*^9}, {3.907493164808448*^9, 3.9074933602943697`*^9}, { 3.90749345376752*^9, 3.90749345376752*^9}, {3.90749362079399*^9, 3.9074936228116426`*^9}, {3.9074936801457877`*^9, 3.907493689572369*^9}, { 3.9074961099100547`*^9, 3.9074961260737057`*^9}, {3.9074964661923075`*^9, 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Estimativas dos \ erros podem ser feitas pelos m\[EAcute]todos discutidos ", ButtonBox["aqui", BaseStyle->"Hyperlink", ButtonData->{ URL["https://arxiv.org/abs/1210.3781"], None}, ButtonNote->"https://arxiv.org/abs/1210.3781"], "." }], "Text", CellMargins-> Dynamic[{{ 0.185 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}], 0.03 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}]}, { 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}], 0.0124074 FrontEnd`AbsoluteCurrentValue[{WindowSize, 2}]}}], CellChangeTimes->{{3.90750554595475*^9, 3.9075056939358053`*^9}, { 3.9075057518050604`*^9, 3.907505751808062*^9}},ExpressionUUID->"acaf6704-a023-4cc8-b5a4-\ 5ad0066ac435"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"40fed88a-8620-47ef-b53c-b50de81b6b11"], Cell["A caminhada aleat\[OAcute]ria", "Section", CellChangeTimes->{{3.9074969600417547`*^9, 3.9074969628891706`*^9}},ExpressionUUID->"fcb07a03-b4f9-4de1-bee7-\ 358c9fc52f03"], Cell["\<\ A integra\[CCedilla]\[ATilde]o pelo m\[EAcute]todo de Monte Carlo \[EAcute] \ muito mais relevante para integrais multidimensionais, mas n\[ATilde]o \ exploraremos isso aqui. (Vejam, por exemplo, a disciplina de M\[EAcute]todos \ Computacionais em F\[IAcute]sica.) Em vez disso, vamos mostrar agora a \ utilidade do m\[EAcute]todo em um outro problema, conectado ao fen\[OHat]meno \ da difus\[ATilde]o de part\[IAcute]culas e ao chamado movimento browniano. \ Trata-se do do modelo da caminhada aleat\[OAcute]ria, cuja an\[AAcute]lise \ vai nos levar a um importante resultado em estat\[IAcute]stica, o teorema do \ limite central.\ \>", "Text", CellChangeTimes->{{3.844004981476076*^9, 3.844005028444255*^9}, { 3.9074967059468217`*^9, 3.9074968790654716`*^9}, {3.907501215345334*^9, 3.9075012221228657`*^9}},ExpressionUUID->"f20b05cf-ded6-4dde-ab38-\ b35759e4ea5b"], Cell["\<\ Considere um processo em que, a cada unidade de tempo, uma part\[IAcute]cula \ movendo-se ao longo de um eixo fixo efetua um passo de comprimento unit\ \[AAcute]rio aleatoriamente no sentido positivo ou no sentido negativo. \ Queremos acompanhar a trajet\[OAcute]ria da part\[IAcute]cula e extrair \ informa\[CCedilla]\[OTilde]es sobre as propriedades estat\[IAcute]sticas \ dessa caminhada. \ \>", "Text", CellChangeTimes->{{3.9074970675125747`*^9, 3.907497223812589*^9}, { 3.9074972571336784`*^9, 3.9074972694295535`*^9}, {3.907501254592318*^9, 3.9075012693544674`*^9}, 3.907502013523644*^9, 3.9075024939566364`*^9, { 3.9075025946728544`*^9, 3.9075026350754967`*^9}},ExpressionUUID->"5b6a37d6-78cd-4611-bb62-\ 20d8a3e465c4"], Cell[TextData[{ "Vamos determinar a trajet\[OAcute]ria ao longo de ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["n", "TI"], TraditionalForm], "errors" -> {}, "input" -> "n", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "ccd11c55-f24d-496b-8957-d8132351b8f3"], " passos, cada um dos quais pode corresponder a um deslocamento ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalDelta]", StyleBox["x", "TI"], "\[Element]", RowBox[{"{", RowBox[{"-1", ",", "+1"}], "}"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\Delta x\\in\\{-1,+1\\}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "e6ac496a-9f60-4e5b-81ba-badc7949ecba"], ". Aqui \[EAcute] muito conveniente a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Accumulate", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Accumulate.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Accumulate.html"], FontFamily->"Source Code Pro", FontWeight->"Normal"], ", que, recebendo como argumento uma lista de passos, gera uma lista de posi\ \[CCedilla]\[OTilde]es sucessivas cujo ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["j", "TI"], TraditionalForm], "errors" -> {}, "input" -> "j", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "e3900634-2db7-4169-925d-4f264cbfae9f"], "-\[EAcute]simo elemento \[EAcute] a soma dos deslocamentos dos ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["j", "TI"], TraditionalForm], "errors" -> {}, "input" -> "j", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "26507c54-6463-4033-b487-4d941a3fc574"], " primeiros passos." }], "Text", CellChangeTimes->{{3.9074970675125747`*^9, 3.907497223812589*^9}, { 3.9074972571336784`*^9, 3.9074972694295535`*^9}, {3.907501254592318*^9, 3.9075012693544674`*^9}, 3.907502013523644*^9, 3.9075024939566364`*^9, { 3.9075025946728544`*^9, 3.907502687104543*^9}, {3.9075027224655323`*^9, 3.9075027367779665`*^9}, {3.9075031155283527`*^9, 3.907503118458233*^9}, { 3.9075031570203433`*^9, 3.9075031623859944`*^9}, 3.907776262951147*^9},ExpressionUUID->"cd828317-1921-41bb-a537-\ 3ccdc2453163"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"n", "=", "1000"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "N\[UAcute]mero", " ", "de", " ", "passos", " ", "da", " ", "caminhada"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"dx", "=", RowBox[{ RowBox[{"2", RowBox[{"RandomInteger", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", "n"}], "]"}]}], "-", "1"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"Lista", " ", "dos", " ", "passos"}], ",", " ", RowBox[{ RowBox[{"cada", " ", "um", " ", "igual", " ", "a"}], " ", "-", RowBox[{"1", " ", "ou"}], " ", "+", "1"}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"pos", "=", RowBox[{"Accumulate", "[", "dx", "]"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Lista", " ", "das", " ", "posi\[CCedilla]\[OTilde]es", " ", "sucessivas", " ", "da", " ", "part\[IAcute]cula"}], " ", "*)"}], " "}], "\[IndentingNewLine]", RowBox[{"ListLinePlot", "[", RowBox[{"pos", ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", RowBox[{"GridLines", "->", "Automatic"}]}], "]"}]}], "Input", CellChangeTimes->{{3.907501533654612*^9, 3.9075016069317303`*^9}, { 3.907503823171193*^9, 3.9075039410339594`*^9}, 3.907508150647882*^9},ExpressionUUID->"ba738baf-0261-4a79-b721-\ 4b71298ea1b1"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"8865992e-5aaa-48b7-babe-ceabf43bf2e3"], Cell["\<\ Para analisarmos o problema estatisticamente, \[EAcute] preciso observar \ muitas trajet\[OAcute]rias diferentes. Vamos produzir as trajet\[OAcute]rias \ de 50 caminhadas diferentes, cada uma com 4000 passos. Para isso, vamos nos \ valer de um la\[CCedilla]o contendo mais de uma instru\[CCedilla]\[ATilde]o \ em cada etapa, o que pode ser feito delimitando essas instru\[CCedilla]\ \[OTilde]es por par\[EHat]nteses:\ \>", "Text", CellChangeTimes->{{3.907503998796731*^9, 3.907504026605747*^9}, { 3.9075040597715287`*^9, 3.907504258049921*^9}, {3.9075082593476753`*^9, 3.907508259774936*^9}, {3.907776296214307*^9, 3.9077763026394167`*^9}},ExpressionUUID->"30bdf238-7cb4-4050-8b27-\ d3628a9bacc0"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"m", "=", "50"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", "4000"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"trajetorias", "=", RowBox[{"{", "}"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Lista", " ", "que", " ", "armazenar\[AAcute]", " ", "as", " ", "trajet\[OAcute]rias"}], " ", "*)"}], RowBox[{"Do", "[", " ", RowBox[{"(*", " ", RowBox[{"La\[CCedilla]o", " ", "sobre", " ", "as", " ", "caminhadas"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"(", "\[IndentingNewLine]", RowBox[{ RowBox[{"dx", "=", RowBox[{ RowBox[{"2", RowBox[{"RandomInteger", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", "n"}], "]"}]}], "-", "1"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Lista", " ", "de", " ", "passos", " ", "de", " ", "uma", " ", "caminhada"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"pos", "=", RowBox[{"Accumulate", "[", "dx", "]"}]}], ";", RowBox[{"(*", " ", RowBox[{ "Lista", " ", "de", " ", "posi\[CCedilla]\[OTilde]es", " ", "sucessivas", " ", "de", " ", "uma", " ", "caminhada"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"AppendTo", "[", RowBox[{"trajetorias", ",", "pos"}], "]"}]}], " ", RowBox[{"(*", " ", RowBox[{ "Agregando", " ", "a", " ", "caminhada", " ", "\[AGrave]", " ", "lista", " ", "de", " ", "trajet\[OAcute]rias"}], " ", "*)"}], "\[IndentingNewLine]", ")"}], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", "m"}], "}"}]}], "\[IndentingNewLine]", "]"}]}]}], "Input", CellChangeTimes->{{3.907504272054891*^9, 3.907504318595047*^9}, 3.9075081724095745`*^9, {3.907743870668929*^9, 3.9077438712019153`*^9}},ExpressionUUID->"4b378f42-c9ce-48e1-8209-\ a9aaa1be20e9"], Cell[TextData[{ "Note que as trajet\[OAcute]rias est\[ATilde]o armazenadas em uma lista de \ listas, equivalente a uma matriz de 50 linhas por 4000 colunas, como podemos \ observar usando a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Dimensions", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Dimensions.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Dimensions.html"], FontFamily->"Source Code Pro", FontWeight->"Normal"], ":" }], "Text", CellChangeTimes->{{3.9075043625412016`*^9, 3.9075044303479004`*^9}, { 3.907504468783676*^9, 3.9075044965781465`*^9}, {3.9075045299771605`*^9, 3.9075045299815917`*^9}},ExpressionUUID->"6eae9bd8-a254-4e41-9f4b-\ e1688c970f2b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Dimensions", "[", "trajetorias", "]"}], " "}]], "Input", CellChangeTimes->{{3.907504431580016*^9, 3.907504450761695*^9}, 3.9075081801092587`*^9},ExpressionUUID->"0c16962b-d29a-4e1b-94c7-\ 98a4dada803f"], Cell["Agora fazemos o gr\[AAcute]fico de todas as trajet\[OAcute]rias:", \ "Text", CellChangeTimes->{{3.9075043625412016`*^9, 3.907504375974493*^9}, { 3.9075045358299437`*^9, 3.9075045409959383`*^9}},ExpressionUUID->"ba4cd318-097f-4b70-91cf-\ 5ca72435087e"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"trajgraf", "=", RowBox[{"ListLinePlot", "[", RowBox[{"trajetorias", ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.9075045534349866`*^9, 3.9075045888111315`*^9}, 3.9075081832745733`*^9},ExpressionUUID->"389eeccc-c86a-4182-a7ee-\ c8d886a6038f"], Cell["\<\ Como esperado, as trajet\[OAcute]rias \[OpenCurlyDoubleQuote]espalham-se\ \[CloseCurlyDoubleQuote] \[AGrave] medida que mais passos s\[ATilde]o dados, \ da mesma forma que a tinta que cai em um copo d\[CloseCurlyQuote]\[AAcute]gua \ se difunde com o tempo. \ \>", "Text", CellChangeTimes->{{3.9075046900988398`*^9, 3.90750475752229*^9}},ExpressionUUID->"fd4acda9-9523-41ed-b9f0-\ 22e3fe67499f"] }, Closed]] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"9ce72fc9-f647-41fe-91d6-414b3b141a87"], Cell["O teorema do limite central", "Section", CellChangeTimes->{{3.9075054612192435`*^9, 3.907505465268289*^9}},ExpressionUUID->"4bcb45ac-a275-4439-9755-\ 795923ec317f"], Cell[TextData[{ "Vamos observar com mais cuidado a estat\[IAcute]stica de posi\[CCedilla]\ \[OTilde]es de muitas caminhadas ap\[OAcute]s um certo n\[UAcute]mero de \ passos. N\[ATilde]o estamos interessados nas trajet\[OAcute]rias em si, mas \ apenas na posi\[CCedilla]\[ATilde]o final, que obtemos da lista de passos com \ o aux\[IAcute]lio da fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Total", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Total.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Total.html"], FontFamily->"Source Code Pro", FontWeight->"Normal"], ". Ap\[OAcute]s obtermos a lista dessas posi\[CCedilla]\[OTilde]es para \ todas as caminhadas, vamos observar o histograma correspondente." }], "Text", CellChangeTimes->{{3.844080785935403*^9, 3.8440808511846857`*^9}, { 3.844081051359322*^9, 3.8440810552837553`*^9}, {3.907505488288866*^9, 3.9075054904596386`*^9}, {3.907505781443448*^9, 3.907505803016143*^9}, { 3.9075058776429853`*^9, 3.9075059128222027`*^9}, {3.907505952830677*^9, 3.907505952833682*^9}, {3.9075060580366526`*^9, 3.907506091082966*^9}},ExpressionUUID->"a3988b43-0ec7-4560-9c00-\ ea5a75203ee3"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"m", "=", "10000"}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", "20000"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"posn", "=", RowBox[{"{", "}"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Lista", " ", "que", " ", "armazenar\[AAcute]", " ", "a", " ", "posi\[CCedilla]\[ATilde]o", " ", "final", " ", "de", " ", "cada", " ", "caminhada"}], " ", "*)"}], RowBox[{"Do", "[", " ", RowBox[{"(*", " ", RowBox[{"La\[CCedilla]o", " ", "sobre", " ", "as", " ", "caminhadas"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"(", "\[IndentingNewLine]", RowBox[{ RowBox[{"dx", "=", RowBox[{ RowBox[{"2", RowBox[{"RandomInteger", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", "n"}], "]"}]}], "-", "1"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Lista", " ", "de", " ", "passos", " ", "de", " ", "uma", " ", "caminhada"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"AppendTo", "[", RowBox[{"posn", ",", RowBox[{"Total", "[", "dx", "]"}]}], "]"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Atualizando", " ", "a", " ", "lista", " ", "que", " ", "cont\[EAcute]m", " ", "as", " ", "posi\[CCedilla]\[OTilde]es", " ", "finais"}], " ", "*)"}], "\[IndentingNewLine]", ")"}], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", "m"}], "}"}]}], "\[IndentingNewLine]", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"histgraf", "=", RowBox[{"Histogram", "[", RowBox[{"posn", ",", RowBox[{"{", "2", "}"}], ",", "\"\\"", ",", "optam", ",", RowBox[{"FrameLabel", "->", RowBox[{"{", RowBox[{"\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ StyleBox[\"x\", \"TI\"], TraditionalForm], \"errors\" -> {}, \"input\" -> \"x\ \", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\"", ",", "\"\<\!\(\*TemplateBox[<|\"boxes\" -> FormBox[ RowBox[{ StyleBox[\"P\", \"TI\"], \"(\", StyleBox[\"x\", \"TI\"], \")\"}], TraditionalForm], \"errors\" -> {}, \"input\ \" -> \"P(x)\", \"state\" -> \"Boxes\"|>,\"TeXAssistantTemplate\"]\)\>\""}], "}"}]}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.9075060244654207`*^9, 3.907506040627218*^9}, { 3.907506097441845*^9, 3.9075061358555255`*^9}, 3.9075082702125435`*^9},ExpressionUUID->"6d64675a-651e-4400-a213-\ 1f6457ef0af0"], Cell[TextData[{ "O histograma lembra bastante aquele de uma distribui\[CCedilla]\[ATilde]o \ gaussiana. De fato, podemos ajust\[AAcute]-lo com uma distribui\[CCedilla]\ \[ATilde]o gaussiana com m\[EAcute]dia ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Mu]", "\[LongEqual]", "0"}], TraditionalForm], "errors" -> {}, "input" -> "\\mu=0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "efcb58e9-e8b2-4bd5-a280-01d6e57e903f"], " e desvio-padr\[ATilde]o ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Sigma]", "\[LongEqual]", SqrtBox[ StyleBox["n", "TI"]]}], TraditionalForm], "errors" -> {}, "input" -> "\\sigma=\\sqrt{n}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "f227fba9-d2b6-43b3-9b97-aaf9c0a282a2"], ":" }], "Text", CellChangeTimes->{{3.907506178269704*^9, 3.9075062511668053`*^9}},ExpressionUUID->"99feb5c0-8809-45b6-910c-\ 8841459ab9c6"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", RowBox[{"histgraf", ",", RowBox[{"Plot", "[", RowBox[{ RowBox[{"PDF", "[", RowBox[{ RowBox[{"NormalDistribution", "[", RowBox[{"0", ",", RowBox[{"Sqrt", "[", "n", "]"}]}], "]"}], ",", "x"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "500"}], ",", "500"}], "}"}]}], "]"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.907506253282384*^9, 3.907506311573513*^9}, 3.9075083110998096`*^9},ExpressionUUID->"cc7e43d9-b11f-4da7-845d-\ 7cf0fa3bc81d"], Cell[TextData[{ "A raz\[ATilde]o para isso est\[AAcute] no ", StyleBox["teorema do limite central", FontWeight->"Bold"], ". Segundo esse teorema, sob condi\[CCedilla]\[OTilde]es bastante gerais, a \ distribui\[CCedilla]\[ATilde]o de probabilidade de uma soma de ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["n", "TI"], "\[GreaterGreater]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "n\\gg 1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "57db4099-7ed0-484a-a52e-93c86ec8cb51"], " vari\[AAcute]veis aleat\[OAcute]rias independentes e identicamente \ distribu\[IAcute]das (com m\[EAcute]dia ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox["\[Mu]", "0"], TraditionalForm], "errors" -> {}, "input" -> "\\mu_0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "82124639-d0a7-43b6-85b9-08e1147d027a"], " e desvio-padr\[ATilde]o ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox["\[Sigma]", "0"], TraditionalForm], "errors" -> {}, "input" -> "\\sigma_0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "4bda63da-b1c0-4a8e-b079-b7f6ca7cbdb5"], ") tende a uma distribui\[CCedilla]\[ATilde]o gaussiana cuja m\[EAcute]dia \ \[EAcute] ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Mu]", "\[LongEqual]", StyleBox["n", "TI"], SubscriptBox["\[Mu]", "0"]}], TraditionalForm], "errors" -> {}, "input" -> "\\mu=n\\mu_0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "fee4c995-461a-492a-bdc0-f13be0fa7f02"], " e cujo desvio-padr\[ATilde]o \[EAcute] ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Sigma]", "\[LongEqual]", SqrtBox[ StyleBox["n", "TI"]], SubscriptBox["\[Sigma]", "0"]}], TraditionalForm], "errors" -> {}, "input" -> "\\sigma=\\sqrt{n}\\sigma_0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "3bc598cd-5fea-4f82-b43a-c88400d6a43d"], ". Em uma caminhada aleat\[OAcute]ria, a posi\[CCedilla]\[ATilde]o depois de \ ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["n", "TI"], TraditionalForm], "errors" -> {}, "input" -> "n", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "b1ecac15-9647-4a84-8ce2-903a0c3085e2"], " passos \[EAcute] justamente dada pela soma" }], "Text", CellChangeTimes->{{3.907506361068646*^9, 3.9075064745265474`*^9}, { 3.9075065124694834`*^9, 3.9075066140586224`*^9}, {3.9075066705924597`*^9, 3.9075066728750477`*^9}, {3.907506709725003*^9, 3.9075067520676775`*^9}, { 3.907506798526366*^9, 3.9075067998262444`*^9}, {3.9075069108220263`*^9, 3.907506911860526*^9}},ExpressionUUID->"5f68e696-6f5a-4ac6-9007-\ 0768518ca0f0"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["x", "TI"], "\[LongEqual]", UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["i", "TI"], "\[LongEqual]", "1"}], StyleBox["n", "TI"], LimitsPositioning -> False], "\[CapitalDelta]", SubscriptBox[ StyleBox["x", "TI"], StyleBox["i", "TI"]]}], TraditionalForm], "errors" -> {}, "input" -> "\\displaystyle x=\\sum_{i=1}^n \\Delta x_i", "state" -> "Boxes"|>, "TeXAssistantTemplate"], ","}], TraditionalForm]],ExpressionUUID->"dfc77c79-7549-4939-8e5d-d63143858695"]], \ "Text", CellChangeTimes->{{3.907506361068646*^9, 3.9075064745265474`*^9}, { 3.9075065124694834`*^9, 3.9075066140586224`*^9}, {3.9075066705924597`*^9, 3.9075066728750477`*^9}, {3.907506709725003*^9, 3.907506756496708*^9}, { 3.907506805226925*^9, 3.9075068124108615`*^9}},ExpressionUUID->"7d0ba220-bc9f-4550-b1d2-\ 0c28f4e3c742"], Cell[TextData[{ "em que ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalDelta]", SubscriptBox[ StyleBox["x", "TI"], StyleBox["i", "TI"]], "\[Element]", RowBox[{"{", RowBox[{"-1", ",", "+1"}], "}"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\Delta x_i\\in\\{-1,+1\\}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "ab43437b-facd-4157-aa09-b482acafc560"], " \[EAcute] uma vari\[AAcute]vel aleat\[OAcute]ria com m\[EAcute]dia ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox["\[Mu]", "0"], "\[LongEqual]", "0"}], TraditionalForm], "errors" -> {}, "input" -> "\\mu_0=0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "64865d53-e05c-47e5-9e68-ae4a2a268258"], " e desvio-padr\[ATilde]o ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox["\[Sigma]", "0"], "\[LongEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "\\sigma_0=1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "bd1202f8-b384-4602-9bef-fa8276222f07"], "." }], "Text", CellChangeTimes->{{3.907506361068646*^9, 3.9075064745265474`*^9}, { 3.9075065124694834`*^9, 3.9075066140586224`*^9}, {3.9075066705924597`*^9, 3.9075066728750477`*^9}, {3.907506709725003*^9, 3.907506756496708*^9}, { 3.907506805226925*^9, 3.9075068773701634`*^9}},ExpressionUUID->"da5f6fc6-de1b-4610-b4d8-\ c5e3e65eb9d7"], Cell[TextData[{ "Em um processo aleat\[OAcute]rio que segue a distribui\[CCedilla]\[ATilde]o \ gaussiana, 95% das vezes o resultado de um experimento desvia-se da \ m\[EAcute]dia por menos de 2 desvios-padr\[ATilde]o. Nosso gr\[AAcute]fico \ das trajet\[OAcute]rias pode ent\[ATilde]o ser bem delimitado pelas curvas ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[PlusMinus]", "2", SqrtBox[ StyleBox["passo", FontSlant -> "Plain"]]}], TraditionalForm], "errors" -> {}, "input" -> "\\pm 2\\sqrt{\\mathrm{passo}}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "bebba728-e16f-4a6c-b843-c72debefece3"], ":" }], "Text", CellChangeTimes->{{3.9075070462676177`*^9, 3.9075072046458716`*^9}},ExpressionUUID->"e03d783a-443c-4d26-99fc-\ c7ba576681be"], Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", RowBox[{"trajgraf", ",", RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"2", "*", RowBox[{"Sqrt", "[", "passo", "]"}]}], ",", RowBox[{ RowBox[{"-", "2"}], "*", RowBox[{"Sqrt", "[", "passo", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"passo", ",", "0", ",", "n"}], "}"}], ",", RowBox[{"PlotStyle", "->", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"Blue", 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