Find $E[\min(V+X, W+X)]$ if $V,X,W$ are not identically distributed

Question

Given that $V,X,W$ are only independent. $V \sim \text{Exp}(\frac 1 4), W \sim \text{Exp}(\frac {3} {20}), X \sim U(-\pi,2-\pi)$

We need to find $E[\min\{V+X,W+X\}]$

Attempt:

Let $Z=\min\{V+X,W+X\}$

$$1-F_Z(z)=P(\min\{V+X,W+X\} >z)=P(V+X > z, W + X > z)$$ by inclusion exclusion principle :

$$=1-P(V+X \le z) - P(W + X \le z) + P(V+X \le z, W+X \le z)$$

and I'm stuck.

Answer 1

Note that $\min(V+X,W+X)=\min(V,W)+X$ therefore by linearity of $E$ all you need is $E(\min(V,W))$ and $E(X)$.

It is explained in this answer for instance that if $V$ and $W$ follow exponential distributions then so does $\min(V,W)$, and the parameters add up.

On the other hand, $E(U(-\pi,2-\pi))=\frac{-\pi+(2-\pi)}{2}=1-\pi$.

Therefore $$E(\min(V+X,W+X))=\frac{1}{\frac 14+\frac 3{20}}+1-\pi$$