Exercício 13

Enunciado

Sejam \(X_1,X_2,\cdots,X_n\) v.a’s independentes tais que \[ P(X_k\leq t)=\begin{cases} 0, & t<0,\\\\ t^k, & 0\leq t < 1, \qquad k=1,2,\cdots,n.\\\\ 1, & t\geq 1. \end{cases} \] Mostre que \[E[max\{X_1,X_2,\cdots,X_n\}]=\binom{n+1}{2}\bigg[1+\binom{n+1}{2}\bigg]^{-1}\]

Prova

Seja \(Z=max\{X_1,X_2,\cdots,X_n\}.\)

\[ \begin{align} F_Z(z)&=P(Z\leq z)\\ &=P(max\{X_1,X_2,\cdots,X_n\} \leq z)\\ &=P(X_1\leq z) P(X_2\leq z)\cdots P(X_n\leq z)\\ &=\begin{cases} 0, & z<0,\\ \prod_{i=1}^{n}z^k, & 0\leq z <1,\\ 1, & z\geq 1. \end{cases} \end{align} \]

Logo, \[ \begin{align} EZ&=\int_{\mathbb{R}}z\;dF_Z\\ &=\int_{0}^{\infty}(1-F_Z(z))dz - \int_{-\infty}^{0}F_Z(z)dz\\ &=\int_{0}^{1}\bigg(1-\prod_{i=1}^{n}z^k\bigg)dz + \int_{1}^{\infty}\big(1-1\big)dz-\int_{-\infty}^{0}0\;dz\\ &=\int_{0}^{1} 1 \;dz - \int_{0}^{1} z^{\sum_{i=1}^{n}k}\;dz\\ &=1- \dfrac{z^{1+\sum_{i=1}^{n}k}}{1+\sum_{i=1}^{n}k}\bigg|_{z\to 0}^{z\to 1}\\ &=1-\dfrac{1}{1+\sum_{i=1}^{n}k}\\ &=1-\frac{1}{1+\frac{n(n+1)}{2}}\\ &=\dfrac{1+\frac{n(n+1)}{2}-1}{1+\frac{n(n+1)}{2}}\\ &=\frac{n(n+1)}{2} \bigg[1+\frac{n(n+1)}{2}\bigg]^{-1}, \end{align} \] em que \(\frac{n(n+1)}{2}=\binom{n+1}{2}\).

Implementação Computacional

N <- 1e6 # tamanho da amostra p/ cada X_i.
n <- 6   # número de v.a`s. 

mat.dados <- matrix(0, nrow = N, ncol= n)

for ( j in 1:n ){
      u <- runif(N,0,1)       # Geracao de Uniforme U(0,1)
      mat.dados[,j]<- u^(1/j) # Inversa Generalizada (gera pseudoaleatorio)
}

Z <- apply(mat.dados, 1, max)


esp.sim <- mean(Z)

esp.teo <- ( n*(n+1)/2 ) * ( 1 + n*(n+1)/2 )^(-1)

list(c(Esp.Teorica = esp.teo, Esp.Simulada = esp.sim))
## [[1]]
##  Esp.Teorica Esp.Simulada 
##    0.9545455    0.9545198