Mean of maximum of exponential random variables (independent but not identical) I am looking for the the mean of the maximum of N independent but not identical exponential random variables. I found the CDF and the pdf but I couldn't compute the integral to find the mean of the maximum. 
The CDF : $ \mathbf{F_{X_{max}}}(x)= \prod_{k=1}^{K} (1-exp(-\lambda_k x)) $ and
The pdf is  : $ \mathbf{f_{X_{max}}}(x)= \sum_{k=1}^{K}\lambda_k exp(-\lambda_k x) \prod_{q=1,q\neq k}^{K} (1-exp(-\lambda_q x)) $
so the mean that I am trying to find is : 
 $ \mathbb{E} [X_{max}] = \int_{0}^{\infty} x \mathbf{f_{X_{max}}}(x) dx $ 
 A: Using this formula for expectation of positive random variables in terms of the survival function and expanding the product in your formula for the cdf,
\begin{align}
EX_\text{max}
  &=\int_0^\infty P(X_\text{max}>x)dx
\\&=\int_0^\infty1-\prod_{i=1}^n(1-e^{-\lambda_i x})dx
\\&=\sum_{S\subseteq\{1,2,\dots,n\}}(-1)^{|S|} \int_0^\infty e^{-x\sum_{j\in S}\lambda_j}dx
\\&=\sum_{S\subseteq\{1,2,\dots,n\}}(-1)^{|S|-1} \frac1{\sum_{j\in S}\lambda_j},
\end{align}
where the outer sum is over all non-empty subsets $S$ of $\{1,2,\dots,n\}$ and $|S|$ denotes the number of elements of $S$.
So for $n=1$, this simplifies to the usual formula
$$
E X_\text{max} = \frac1{\lambda_1},
$$
for $n=2$, 
$$
E X_\text{max} = \frac1{\lambda_1} + \frac1{\lambda_2} - \frac1{\lambda_1+\lambda_2},
$$
for $n=3$,
$$
E X_\text{max} = \frac1{\lambda_1} + \frac1{\lambda_2} + \frac1{\lambda_3} - \frac1{\lambda_1+\lambda_2} - \frac1{\lambda_1+\lambda_3} - \frac1{\lambda_2+\lambda_3} + \frac1{\lambda_1+\lambda_2+\lambda_3},
$$
and so on.
