[Math] Let $X\sim\mathrm{Exp}(1)$ and $Y\sim\mathrm{Exp}(2)$ independent random variables. Let $Z = \max(X, Y)$. Calculate expected value of $Z$.

Question

Here a question:

Let $X \sim \mathrm{Exp}(1)$ and $Y\sim\mathrm{Exp}(2)$ be independent random variables. Let $Z = \max(X, Y)$. Calculate expected value of $Z$.

My try:

\begin{align}
P(Z \le z) & = P(\max(X, Y) \le z) = 1 – P(\max(X, Y) > z) \\[10pt]
& = 1-P(X > z)P(Y > z) = 1 – (e^{-z} \cdot e^{-2z}) = 1 – e^{-3z}.
\end{align}

find distribution function: $F_z = \frac{d}{dz} \space 1 – e^{-3z} = 3e^{-3z}. $

Find expected value: $\int_0^\infty z\cdot F_z \,dz = 3\int_0^\infty z\cdot e^{-3z}\,dz = \frac{1}{3}$.

Right solution is $\frac{7}{6}$. Whats wrong in my way?

Answer 1

We have $\Pr(Z\le z)=\Pr((X\le z)\cap (Y\le z))=(1-e^{-z})(1-e^{-2z})$ (for $z\gt 0$.) Differentiate to find the density, and then calculate the expectation as usual.