basic_statistics.jl

### A Pluto.jl notebook ###
# v0.19.5

using Markdown
using InteractiveUtils

# ╔═╡ 653103ae-a91b-11ec-3cbd-6bcb6a71d48a
begin
	using PlutoUI, Plots, StatsBase, Measures, Distributions, QuadGK
	using LaTeXStrings, Random
end

# ╔═╡ 26e4377a-377d-47b9-9305-6b210d8b482b
md"""
# Estatística na Linguagem Julia
Version 0.0.1
## Probabilidade
_Probabilidade_ é a chance de algum evento acontecer, quantificados como números entre 0 e 1, com valores mais altos indicando uma maior probabilidade de ocorrência.
Uma maneira de representar a idéia acima é considerar uma _espaço de probabilidade_ que consiste de 3 elementos:

1 - **Espaço Amostral**: O conjunto de todas os possíveis resultados de experimentos.\
O espaço amostral é representado por **ômega** Ω. Considere como exemplo jogar uma moeda 2 vezes onde _a-cara_, _b-coroa_. O espaço amostral é o conjunto com todos os resultados possíveis:\
Ω = {aa, ab, ba, bb}\
2 - **Coleção de Eventos**: Cada evento é um subconjunto do _espaço amostral_ Ω.\
O evento A de resultados com pelo menos uma cara. A = {aa, ab, ba}\
3 - **Função de Probabilidade**: Indica a chance de cada evento ocorrer.\
$\mathbb{P(A) = A/Ω}$
## Experimentos Aleatórios
"""

# ╔═╡ 1952bd5c-162e-4f32-9e7c-3eb5a8c9f7b1
md"""
### Urna com Bolas Coloridas
Considere uma urna com 20 bolas. Sendo 5 delas azuis e 15 vermelhas. Qual a probabilidade de se retirar uma bola com a cor azul?\
**Ω = {a, a, a, a, a, v, v, v, v, v, v, v, v, v, v, v, v, v, v, v}**\
**A = {a, a, a, a, a}**\
$\mathbb{P(A) = A/Ω}$
"""

# ╔═╡ 8d2f266a-caf8-41a3-ae7a-5b4bae8e4c78
function ProbabilityAOmega()
	Ω = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
	A = [0, 0, 0, 0, 0];
	p = length(A)/length(Ω);
	println("A probabilidade de se retirar uma bola azul é $p");
end

# ╔═╡ 5e8ad639-4cc4-45d0-a409-9ae2cbc932c4
ProbabilityAOmega()

# ╔═╡ c74fa1b1-5511-42c1-bf0a-5e97132e0f02
md"""
### Simulação Monte Carlo
A _Simulação de Monte Carlo_, também conhecida como _Método de Monte Carlo_ ou uma _simulação de probabilidade múltipla_, é uma técnica matemática, que é usada para estimar os possíveis resultados de um evento incerto. O Método de Monte Carlo foi inventado por John von Neumann e Stanislaw Ulam durante a Segunda Guerra Mundial para melhorar a tomada de decisão em condições incertas. Foi nomeado em homenagem a uma conhecida cidade de cassinos, chamada Mônaco, uma vez que o acaso é principal elemento da abordagem de modelagem, semelhante a um jogo de roleta. Experimentos com Monte Carlo se baseiam na _Lei dos Grandes Números_. Este conjunto de afirmações matemáticas afirma que as **médias empíricas** convergem para os valores esperados.

$$\overline{X}_n = \frac{1}n \sum_{k=1}^n X_k,$$

Simularemos o exemplo da urna com 20 bolas abaixo:
"""

# ╔═╡ 3ef272d5-7333-445d-8b17-df578a5e0576
begin
	# Bolas azuis - 0 Bolas vermelhas - 1
	Ω = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
	A = [0, 0, 0, 0, 0]; # Evento A representa o evento de se retirar bolas azuis
	NB = 10 ^ 6; # Número de experimentos - 1mi
	# Observamos no resultado com 1mi experimentos aleatórios que
	# se aproxima do valor numérico A/Ω - 0.25
	resultado = sum([(sample(Ω) in A) for _ in 1:NB]) / NB; # Média empírica
	println("Estimativa do método de monte carlo: $resultado");
end

# ╔═╡ 6083d9ef-78f6-4aa3-a1b4-2079186db166
md"""
## Trabalhando com Conjuntos
Uma parte importante da probabilidade são os conjuntos. É através deles que representamos eventos do espaço amostral, por calcular diferenças, interceções e uniões.
"""

# ╔═╡ 57055390-7986-4b1e-a912-a7694774144b
function WorkingWithSets()
	C = Set([2, 7, 2, 3]);
	D = Set(1:6);
	omega = Set(1:10);
	cUnionD = union(C, D);
	cIntersectionD = intersect(C, D)
	cDifferenceD = setdiff(D, C);
	dComplement = setdiff(omega, D);
	dSubsetOmega = issubset(D, omega);

	println("Ômega: $omega");
	println("C: $C");
	println("D: $D");
	println("C ∪ D: $cUnionD");
	println("C ⋂ D: $cIntersectionD");
	println("D diferença C: $cDifferenceD");
	println("D complemento: $dComplement");
	println("D é subconjunto de Ômega: $dSubsetOmega");
end

# ╔═╡ 52564831-4087-4cec-b47c-f88e6a045576
WorkingWithSets()

# ╔═╡ 2176a0a5-65f3-4434-a480-d6aea8adb68b
md"""
## A Probabilidade da União
"""

# ╔═╡ 15126370-335c-4cdb-8253-a71e4a1d31d3
md"""
# Variáveis Aleatórias
Uma _variável aleatória_ X é uma **função do espaço amostral Ω**, e pode ser representada tanto em números inteiros, reais ou até mesmo em um vetor de valores.

Variáveis aleatórias podem ser **discretas ou contínuas**.
_Variáveis aleatórias discretas_ têm valores finitos, enquanto _variáveis aleatórias contínuas_ têm valores que não podem ser mensurados.

Como um exemplo, considere um espaço amostral Ω que consiste de seis nomes. Assuma que a função de probabilidade(ou medida de probabilidade), $\mathbb{P(.)}$ atribui probabilidade uniforme para cada um dos nomes. Deixe **X: Ω -> $\mathbb{Z}$**, ser a função(i. e variável aleatória) que conta o número de letras em cada nome. A questão é então encontrar: **p(x):=$\mathbb{P}$(X = x), for  x ∈ $\mathbb{Z}$**.

A função p(x) representa a _distribuição de probabilidade_ da variável aleatória X. Nesse caso, como X mede a quantidade de caracteres dos nomes, X é uma variável aleatória discreta, e sua distribuição de probabilidade pode ser representada por uma _Função Massa de Probabilidade(PMF)_ como p(x).
"""

# ╔═╡ 60e0db2c-7631-48cc-b770-1c32aa829fd5
begin
	names = ["Mary", "Mel", "David", "John", "Kayley", "Anderson"];
	randomName() = rand(names);
	X = 3:8;
	N = 10 ^ 6;
	sampleLengths = [length(randomName()) for _ in 1:N];
	bar(X, counts(sampleLengths) / N, ylims=(0, 0.35), xlabel="Name length", ylabel="Estimated p(x)", legend=:none, c=:blue)
end

# ╔═╡ 014073e2-2866-435a-92b6-1e6eb1b6a822
md"""
Gráfico da distribuição de probabilidade de um dado de 6 lados honesto.
"""

# ╔═╡ 33dcde77-1bff-411f-9639-cbe4329bd490
begin
	dado = [1, 2, 3, 4, 5, 6];
	DN = 10 ^ 6;
	DX = 1:6;
	amostras = [rand(dado) for _ in 1:N];
	bar(DX, counts(amostras) / DN, ylims=(0, 0.2), title="Distribuição de Prob. de Dado de 6 Faces Honesto", xlabel="Face do dado", ylabel="Probabilidade p(x)", legend=:none, c=:blue)
end

# ╔═╡ d381e4ca-1cf9-4f3e-8158-ca4fc3e908da
md"""
### Três Exemplos diferentes de Distribuições de Probabilidades
"""

# ╔═╡ 4d1c30df-eb95-45cf-bbc8-b34c834511d7
function ExecuteDistributions()
	pDiscrete = [.25, .25, .5];
	xGridD = 0:2;
	pContinuous(x) = 3/4*(1 - x ^ 2);
	xGridC = -1:.01:1;
	pContinuous2(x) = x < 0 ? x+1 : 1-x;
	p1 = plot(xGridD, line=:stem, title="A", pDiscrete, marker=:circle, c=:blue, ms=6, msw=0);
	p2 = plot(xGridC, pContinuous.(xGridC), c=:blue, title="B");
	p3 = plot(xGridC, pContinuous2.(xGridC), c=:blue, title="C");
	plot(p1, p2, p3, layout=(1, 3), legend=false, ylims=(0, 1.1), xlabel="x", ylabel=["Probabilidade" "Densidade" "Densidade"], size=(1200, 400), margin=5mm)
end

# ╔═╡ 6f930a21-bb44-47ac-8e7a-f82f0bf4d617
ExecuteDistributions()

# ╔═╡ c925cee6-f0de-4455-97a1-86298ab4d0cd
md"""
## Média ou Esperança de Variáveis Aleatórias Discretas
Este exemplo calcula a média para o exemplo A no exercício anterior.\
$$\sum_{x} xp(x)$$
"""

# ╔═╡ cc11e376-6a80-4bd6-9192-24645c2f38f6
function DiscreteMeanPx()
	x = 0:2;
	pDiscrete = [.25, .25, .5];
	mean = sum(x .* pDiscrete);
	md"A média para a variável discreta p(x) A é **$mean**" 
end

# ╔═╡ 40883644-b558-42c6-bc40-73bf32ec8a20
DiscreteMeanPx()

# ╔═╡ 3656ac99-54c9-4489-8065-bd23a821105a
md"""
## Média ou Esperança de Variáveis Contínuas
Médias baseadas nas variáveis B e C dos três exemplos anteriores. Via integração numérica.
$$\int_{-\infty}^\infty xf(x)dx$$

"""

# ╔═╡ 48173688-6f1d-40cd-9221-db9f8499b449
function ContinuousMeanBC()
	sup = (-1, -1);
	fa(x) = 3/4*(1-x^2);
	fb(x) = x < 0 ? x + 1 : 1 - x;

	expect(f, support) = quadgk((x) -> x * f(x), support...)[1];
	println("Mean fa: ", expect(fa, sup), ", Mean fb: ", expect(fb, sup))
end

# ╔═╡ ab392208-b361-4a70-98b0-3d5bfe79a9d6
ContinuousMeanBC()

# ╔═╡ 35f01290-5202-4822-bf78-0972c6f1c9d3
md"""
## Mostrando que a variância de p(x) é igual a média de y
Mostrando que a variância de p(x) é igual a média de y em variáveis contínuas.
"""

# ╔═╡ f665d69a-c231-4754-a9f4-ef5ea5d5fb08
function VarianceXEqualsMeanY()
	dist = TriangularDist(4, 6, 5);
	N = 10 ^ 6;
	data = rand(dist, N);
	yData = (data .- 5).^2;
	println("Média: ", mean(yData), " Variância: ", var(data));
	p1 = histogram(data, xlabel="x", bins=80, normed=true, ylims = (0, 1.1));
	p2 = histogram(yData, xlabel="y", bins=80, normed=true, ylims = (0, 15));
	plot(p1, p2, ylabel = "Proporção", size=(800, 450), legend=:none)
end

# ╔═╡ d3ec82a5-0e9a-46f6-942c-07c194dc5abd
VarianceXEqualsMeanY()

# ╔═╡ 04d3b996-6ed5-45f1-9764-19cc15efe2de
md"""
## Funções que Descrevem Distribuições
Além das **pmf(probability mass function)** e **pdf(probability density function)** existem outras funções representantes de distribuições de probabilidade como a _Função Distribuição Cumulativa (cdf)_, _Função Distribuição Cumulativa Complementar (ccdf)_ e _Função Distribuição Cumulativa Inversa (icdf)_. Distribuições de probabilidade podem ser descritas de várias formas alternativas.  
## CDF a partir do Somatório de Riemman de uma PDF
"""

# ╔═╡ 8c5c3321-55dc-40af-aec5-0146a2c00238
function CDFFromRiemmanPDF()
	a, b = -1.5, 1.5;
	f2(x) = (x < 0 ? x + 1 : 1 - x) * (abs(x) < 1 ? 1 : 0);
	delta = 0.01;

	xGrid = a:delta:b;
	F(x) = sum([f2(u) * delta for u in a:delta:x]);
	y = [F(u) for u in xGrid];
	plot(xGrid, y, c=:blue, xlims=(a, b), ylims=(0, 1), xlabel=L"x", ylabel=L"F(x)", legend=:none)
end

# ╔═╡ 2ac8ef95-02a6-4bc7-97a4-92f8b148b55e
CDFFromRiemmanPDF()

# ╔═╡ a8e2dfb7-8251-466c-9b97-09f82bdd9c30
md"""
# Pacotes utilizados para lidar com distribuições
Abaixo um exemplo que mostra como criar amostras a partir de um WeightVector(vetor de pesos). As probabilidades das amostras da variável data são baseadas nos respectivos pesos.
"""

# ╔═╡ 2e458ad2-6536-4f4d-aa0d-24676d1f9ff9
function ExampleWeightVector()
	Random.seed!(1)
	grade = ["A", "B", "C", "D", "E"]
	weightVect = Weights([.1, .2, .1, .2, .4])
	N = 10^6
	data = sample(grade, weightVect, N)
	data = [count(i -> (i == g), data) for g in grade]/N
	println(data)
end

# ╔═╡ ac25ff1f-4a91-4135-a678-30d7e58deeba
ExampleWeightVector()

# ╔═╡ 0c98a2db-4f43-461f-8314-eb0cdc919567
md"""
## Usando Objetos para criar distribuições
Exemplo mostra como usar o pacote _Distributions.jl_ para criar uma distribuição triangular a partir da seguinte densidade: \

$$f(x)=\begin{cases} x & \quad \text{para } x \in [0,1], \\ 2 - x & \quad \text{para } x \in (1,2]. \end{cases}$$
"""

# ╔═╡ e9ae5123-84cb-4000-9f74-74ef3cd19b5a
function GenerateTriagularDistribution()
	dist = TriangularDist(0, 2, 1)
	xGrid = 0:.01:2
	uGrid = 0:.01:1
	p1 = plot(xGrid, pdf.(dist, xGrid), c=:blue, xlims=(0, 2), ylims=(0, 1.1), xlabel="x", ylabel="f(x) ")

	p2 = plot(xGrid, cdf.(dist, xGrid), c=:blue, xlims=(0, 2), ylims=(0, 1.1), xlabel="x", ylabel="F(x) ")

	p3 = plot(uGrid, quantile.(dist, uGrid), c=:blue, xlims=(0, 1), ylims=(0, 2), xlabel="u", ylabel=L"F^{-1} (u) ")

	plot(p1, p2, p3, legend=false, layout=(1, 3), size=(600, 280))
end

# ╔═╡ ed4959f6-74ff-40d9-b37c-c413b165b679
GenerateTriagularDistribution()

# ╔═╡ bcb51ed2-c629-41b1-bd70-f44e6c26ed51
md"""
# Descritores de distribuições
Como recuperar ou modificar parametros de distribuições.
"""

# ╔═╡ 1668be62-eaee-4035-8f22-5c430bb5c0b4
function DistributionParams()
	dist = TriangularDist(0, 2, 1)
	println("Parâmetros: \t\t\t", params(dist))
	println("Descritores Centrais: \t\t", mean(dist), "\t", median(dist))
	println("Descritores de Dispersão: \t", var(dist), "\t", std(dist))
	println("Descritores de Momento de Alta Ordem: \t", skewness(dist), "\t", kurtosis(dist))
	println("Intervalo: \t\t\t\t", minimum(dist), "\t", maximum(dist))
	println("Moda: \t\t\t\t", mode(dist), "\tModas: ", modes(dist))
end

# ╔═╡ 2f268499-d2b7-49d3-9f04-36968b7a38b7
DistributionParams()

# ╔═╡ 5dfb3e1e-d185-4c79-a6a6-a9e3c89c7c18
md"""
## Usando a função rand() com Distribuições
Neste exemplo geramos observações aleatórias de um objeto pela função rand() e comparamos as médias.
"""

# ╔═╡ bc85308b-cedd-4c42-bb89-dee7cb9aa171
function UsingRand()
	Random.seed!(1)
	dist1 = TriangularDist(0, 10, 5)
	dist2 = DiscreteUniform(1, 5)
	theorMean1, theorMean2 = mean(dist1), mean(dist2)
	
	N = 10 ^ 6
	data1 = rand(dist1, N)
	data2 = rand(dist2, N)
	estMean1, estMean2 = mean(data1), mean(data2)
	println("Distribuição Triangular Simétrica no intervalo [0, 10] tem média $theorMean1. Estimada: $estMean1")
	println("Distribuição Discreta Uniforme no conjunto {1, 2, 3, 4, 5} tem media $theorMean2. Estimada: $estMean2")
end

# ╔═╡ 52c26cec-849c-45e4-badb-101269bccf4c
UsingRand()

# ╔═╡ e02eb0b9-68b7-460e-bed2-bf7231e90df1
md"""
## Transformada Inversa
"""

# ╔═╡ a3cb8a4a-855d-4bf5-8a15-2a8e64b37333
function InverseTransform()
	triangDist = TriangularDist(0, 2, 1)
	xGrid = 0:.1:2
	N = 10 ^ 6
	inverseSampleData = quantile.(triangDist, rand(N))
	histogram(inverseSampleData, bins=40, normed=true, ylims=(0, 1.1), label="Transformada Inversa - Dado")
	plot!(xGrid, pdf.(triangDist, xGrid), c=:red, lw=4, xlabel="x", label="PDF", ylabel="Density", legend=:topright)
end

# ╔═╡ f013624d-222f-437c-a8f9-2998a7db9d7c
InverseTransform()

# ╔═╡ 810475e3-af18-48bd-ac5a-824635e13ce3
md"""
# Famílias de Distribuições Discretas
"""

# ╔═╡ 5d69b014-9269-42f8-a487-db7da7af097a
function GenerateDiscreteVariableFamily()
	dists = [
		DiscreteUniform(10, 20),
		Binomial(10, .5),
		Geometric(0.5),
		NegativeBinomial(10, .5),
		Hypergeometric(30, 40, 10),
		Poisson(5.5)
	]
	println("Distribuição \t\t\t\t\t\t\t\t\t Parâmetros \t Suporte")
	data = reshape([dists; params.(dists); ((d)->(minimum(d), maximum(d))).(dists)], length(dists), 3);
	println(data[1, 1], " \t\t ", data[1, 2], " \t\t ", data[1, 3])
	println(data[2, 1], " \t ", data[1, 2], " \t\t ", data[1, 3])
	println(data[3, 1], " \t\t ", data[1, 2], " \t\t ", data[1, 3])
	println(data[4, 1], " \t\t ", data[1, 2], " \t\t ", data[1, 3])
	println(data[5, 1], " ", data[1, 2], " \t\t ", data[1, 3])
	println(data[6, 1], " \t\t\t ", data[1, 2], " \t\t ", data[1, 3])
end

# ╔═╡ c87e9dd8-589f-4800-83de-95a4c3e410fd
GenerateDiscreteVariableFamily()

# ╔═╡ 2234e94b-818b-4a80-8446-3ba702f63062
md"""
## Distribuição Uniforme
A _Distribuição Uniforme Discreta_ é uma distribuição que aplica igual probabilidade a todos os resultados dos eventos.\
No exemplo abaixo demonstramos a distribuição _Uniforme Discreta_.\
$$\mathbb{P}(X=x)=\frac{1}{6} \quad \text{para } x = 1,...,6.$$
"""

# ╔═╡ 5ec6a595-c99e-40e3-a952-9cbc05bc2451
function DieToosDiscreteUniform()
	faces, N = 1:6, 10 ^ 6
	mcEstimate = counts(rand(faces, N), faces) / N

	println("Média: $(sum(mcEstimate .* collect(1:6)))")
	plot(faces, mcEstimate, line=:stem, marker=:circle, c=:blue, ms=10, msw=0, lw=4, label="MC estimatimativa")
	plot!([i for i in faces], [1/6 for _ in faces], line=:stem, marker=:xcross, c=:red, ms=6, msw=0, lw=2, label="PMF", xlabel="Face", ylabel="Probabilidade", ylims=(0, 0.22))
end

# ╔═╡ 821f213b-e7a8-4cf9-b732-ac5c54ce8f57
DieToosDiscreteUniform()

# ╔═╡ 7ed13adf-d18b-436f-9301-aa31bb900e3d
md"""
## Distribuição Binomial
A Distribuição Binomial Discreta é uma distribuição que aparece quando independentes, múltiplas e idênticas tentativas booleanas(true/false, yes/no...) são executadas.
O exemplo abaixo simula o giro de uma moeda de dois lados justa, n vezes. Se a probabilidade de obter cara num único giro é p, então a probabilidade de obtermos x caras é dada pela equação abaixo:\
\
$$\mathbb{P} = (X = x) =\binom{n}{x}p^x(1-p)^{n-x} \quad \text{para } x = 0, 1,...,n$$
"""

# ╔═╡ e8383bfa-14a4-45ca-8be9-71f237d75486
function GenerateBinomialDistribution()
	binomialRV(n, p) = sum(rand(n) .< p)
	p, n, N = .5, 10, 10 ^ 6
	bDist = Binomial(n, p)
	xGrid = 0:n
	bPmf = [pdf(bDist, i) for i in xGrid]
	data = [binomialRV(n, p) for _ in 1:N]
	pmfEst = counts(data, 0:n) / N
	
	total = sum(bPmf)
	println("Total da função massa de probabilidade: $total")
	println("Média: $(sum(xGrid .* bPmf))")
	println("Média: $(mean(bDist))")

	plot(xGrid, pmfEst, line=:stem, mark=:circle, c=:blue, ms=10, msw=0, lw=4, label="MC Estimativa")

	plot!(xGrid, bPmf, line=:stem, marker=:xcross, c=:red, ms=6, msw=0, lw=2, label="PMF", xticks=(0:1:10), ylims=(0, .3), xlabel="x", ylabel="Probabilidade")
end

# ╔═╡ 2c07d0cc-44da-48a8-8b59-b001f40cd8c6
GenerateBinomialDistribution()

# ╔═╡ 479bdfbf-bf61-4460-9b8a-30db48552bd4
md"""
## Distribuição Gemométrica
$$\mathbb{P}(X=x)=p(1-p)^{x-1} \quad \text{para } x = 1,2,....$$
$$\mathbb{P}(\tilde{X}=x)=p(1-p)^x \quad \text{para } x = 0,1,2,....$$
"""

# ╔═╡ c18fd297-8c6c-4cfa-bc64-0b4d9e066d57
function PlayRoulette()
	function rouletteSpins(p)
		x = 0
		while true
			x +=1
			if rand() < p
				return x
			end
		end
	end

	p, xGrid, N = 18/37, 1:7, 10^6
	gDist = Geometric(p)
	gPmf = [pdf(gDist, x-1) for x in xGrid]

	mcEstimate = counts([rouletteSpins(p) for _ in 1:N], xGrid)/N

	println("Média: $(sum(mcEstimate .* xGrid)) ?")
	println("Média: $(mean(gDist))")
	println("Total: $(sum(mcEstimate))")
	
	plot(xGrid, mcEstimate, line=:stem, marker=:circle, c=:blue, ms=10, msw=0, lw=4, label="MC Estimativa")
	
	plot!(xGrid, gPmf, line=:stem, marker=:xcross, c=:red, ms=6, msw=0, lw=2, label="PMF", ylims=(0, .5), xlabel="x", ylabel="Probabilidade")
end

# ╔═╡ fbdb992f-971b-4063-8c60-3dcd78a91607
PlayRoulette()

# ╔═╡ f82526dd-8983-442a-8321-c9ae7a5ea940
md"""
## Distribuição Binomial Negativa
$$\mathbb{P}(X=x)=\binom{x-1}{r-1}p^{r}(1-p)^{x-r} \quad \text{para } x=r,r+1,r+2,....$$

$$\mathbb{P}(\tilde{X}=x)=\binom{x + r -1}{x}p^{r}(1-p)^x \quad \text{para } x=0,1,2,....$$
"""

# ╔═╡ 11595fb9-ba58-4435-852c-31537fe1e017
function PlayRouletteInNegativeBinomial()
	function rouletteSpins(r, p)
		x = 0
		wins = 0;
		while true
			x += 1
			if rand() < p
				wins += 1
				if wins == r
					return x
				end
			end
		end
	end
	r, p, N = 5, 18/37, 10^6
	xGrid = r:r+15
	mcEstimate = counts([rouletteSpins(r, p) for _ in 1:N], xGrid)/N

	nbDist = NegativeBinomial(r, p)
	nbPmf = [pdf(nbDist, x-r) for x in xGrid]

	println("Total da massa de probabilidade $(sum(mcEstimate))")
	media = sum(xGrid .* mcEstimate)
	println("Média: $media")
	println("Média da biblioteca: $(sum(nbPmf .* xGrid))")

	plot(xGrid, mcEstimate, line=:stem, marker=:circle, c=:blue, ms=10, msw=0, lw=4, label="MC estimativa")
	plot!(xGrid, nbPmf, line=:stem, marker=:xcross, c=:red, ms=6, msw=0, lw=2, label="PMF", xlims=(0, maximum(xGrid)), ylims=(0, .2), xlabel="x", ylabel="Probabilidade")
end

# ╔═╡ 658d6d3c-1492-4c1f-b285-0898becff342
PlayRouletteInNegativeBinomial()

# ╔═╡ 4743106c-5dbe-468c-b254-1789d71974f4
md"""
## Distribuição Hipergeométrica
$$p(x)=\frac{\binom{K}{x}\binom{L-K}{n -x}}{\binom{L}{n}} \quad \text{para } x = max(0, n + K - L),...,min(n, K)$$
"""

# ╔═╡ ef39ca7e-4aae-403a-9074-ee6dcb725d06
function GenerateHyperGeometricDistribution()
	L, K, n = 500, [450, 400, 250, 100, 50], 30
	hyperDists = [Hypergeometric(k, L-k, n) for k in K]
	xGrid = 0:1:n
	pmfs = [pdf.(dist, xGrid) for dist in hyperDists]
	labels = "Successos = " .* string.(K)

	bar(xGrid, pmfs, alpha=0.6, c=[:orange :purple :green :red :blue], label=hcat(labels...), ylims = (0, .25), xlabel="x", ylabel="Probabilidade", legend=:top)
end

# ╔═╡ eed06908-74b2-49aa-a650-928dfa6e3aac
GenerateHyperGeometricDistribution()

# ╔═╡ c2bb2537-68be-428b-9263-5a5b126c99a3
md"""
## Distribuição Poisson
$$\prod_{i=1}^NU_i\geq e^{-\lambda} > \prod_{i=1}^{N+1}U_{i,}$$
"""

# ╔═╡ ba5f4f0f-43bb-45d4-86c7-17d406a8da97
function GeneratePoissonDistribution()
	function prn(lambda)
		k, p = 0, 1
		while p > MathConstants.e^(-lambda)
			k += 1
			p *= rand()
		end
		return k - 1
	end

	xGrid, lambda, N = 0:16, 5.5, 10 ^ 6
	pDist = Poisson(lambda)
	bPmf = pdf.(pDist, xGrid)
	data = counts([prn(lambda) for _ in 1:N], xGrid) / N
	plot(xGrid, data, line=:stem, marker=:circle, c=:blue, ms=10, msw=0, lw=4, label="Mc Estimate")
	plot!(xGrid, bPmf, line=:stem, marker=:xcross, c=:red, ms=6, msw=0, lw=2, label="PMF", ylims=(0,0.2), xlabel="x", ylabel="Probability of x events")
		
end

# ╔═╡ 5a444ede-f21a-4b78-a007-f0826ef5b9a2
GeneratePoissonDistribution()

# ╔═╡ 249458e1-a531-4d86-90ea-bab1f73b2400
md"""
$$\sum_{k=1}^n+\prod_{k=2}^n+\lim_{x=\infty}f(x)=0$$
$$k <= x$$
"""

# ╔═╡ fa86eb5d-6b26-49b4-a4c3-ba4bed42e102
begin
	NSAMP = 10 ^ 6
	println("Amostragem uniforme pela função rand(N). Padrão")
	uniform = rand(N)
	histogram(uniform, bins=20, normed = true, legend=:false)
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
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[compat]
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QuadGK = "~2.4.2"
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"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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