I have some problems about this question.
$X$ and $Y$ are two independent random variables: X is an exponential
random variable, Y is a uniform random variable over $[0, a]$. Given
that $EX = EY = 6$, find the density function of the random variable
$T = max(X, Y ).$
Now, I found that $a=12$, $f(x)=1/12$ when $0 < x < 12$ ($0$ otherwise), and $\beta = 6$. I solved some problems similar to this but the distributions were not different. So honestly I do not know what to do to find the density function. Any help is appreciated.
Best Answer
Hint: Let $F_{Z}$ be the cumulative distribution of a random variable $Z$. If $X$ and $Y$ are independent random variables, then \begin{multline*} F_{T}(t)=\mathbb{P}\{T\leq t\}=\mathbb{P}\{\max\{X,Y\}\leq t\}=\mathbb{P}(\{X\leq t\}\cap\{Y\leq t\})=\mathbb{P}\{X \leq t\} \mathbb{P} \{ Y \leq t \}\\=F_{X}(t)F_{Y}(t). \end{multline*}