Let $X\sim\mathrm{Exp}(1),Y\sim\mathrm{Exp}(2)$ independent random variables. Let $Z = \min(X, Y)$. Calculate $E(Z^2)$.

Exercise:

Let $X\sim\mathrm{Exp}(1)$, $Y\sim\mathrm{Exp}(2)$ independent random variables. Let $Z = \min(X, Y)$. Calculate $E(Z^2)$.

What is the approach for such question? What is the way to solve it?

• Unfortunately there are two distinct conventional meanings of $Y\sim\mathrm{Exp}(2)$: it can mean the expected value is $2$, or that the intensity is $2$, so that the expected value is $1/2$. Which did you intend here? May 13 '15 at 22:42
• I understand that very well, but it doesn't address the question. Did you mean an exponentially distributed random variable with expected value $2$ or one with expected value $1/2$? May 13 '15 at 23:05
• @MichaelHardy A random variable with density $f_X(t)=2e^{-2t}$. May 13 '15 at 23:23
Hint: $P(Z \ge z) = P(X \ge z) P(Y \ge z)$. Find the distribution of $Z$.