Let $X_1,…,X_n$ be independent exponential random variables with parameter $\gamma$. Let $Y=\max(X_1,…,X_n)$. Find density $f_Y(y)$

Question

Let $X_1,…,X_n$ be independent exponential random variables with parameter $\gamma$. Let $Y=\max(X_1,…,X_n)$. Find density $f_Y(y)$

So I believe I do something like this:

$f_Y(y)=\frac{d}{dy}F_Y(y)=\frac{d}{dy}P(Y\leq y)=\frac{d}{dy}P(\max(X_1,…,X_n)\leq y)$

Am I doing this right? And how to continue from here? Not sure how to deal with the max function.

Answer 1

$$P(Y \leq y) = P(\max (X_1, ..., X_n) \leq y)$$ $$ = P( \{ X_1 \leq y \} \cap \{ X_2 \leq y \} \cap \cdots \cap \{ X_n \leq y \})$$

$$ = \prod_{i=1}^n P \left ( X_i \leq y \right )$$

$$ = \left (1 - e^{ - \gamma y} \right )^n $$

$$f_Y(y) = \begin{cases}n \left ( 1 - e^{ - \gamma y} \right )^{n-1} \gamma e^{- \gamma y} & y \geq 0\\ 0 & y < 0 \end{cases} $$