Question

Suppose that X1,...,Xn form a random sample of size n from a continuous distribution with the following pdf. f(x)=2x for 0 < x < 1. Let Y=max{X1,...,Xn} Evaluate E[y]

Answer 1

Find the distribution function of \(Y\):
\[\begin{align*}F_Y(x)&=\prod_{k=1}^nF_X(x_k)\\
&=\prod_{k=1}^n\int_0^x f(t_k)\,dt_k&\text{where }f(t_k)\text{ denotes the density of }X_k\\
&=\prod_{k=1}^nx^2\\
&=\begin{cases}0&\text{for }x<0\\x^{2n}&\text{for }0<x<1\\1&\text{for }x>1\end{cases}\end{align*}\]
Take the derivative to get the density function \(f_Y(x)\), then compute the integral,
\[E(Y)=\int_{-\infty}^\infty xf_Y(x)\,dx\]