Exponential distribution Max(X,Y)

Question

This problem appeared in a past Quiz. I'm still not able to solve it:

Note(λ = 1/B). Suppose X~exponential(λ = 1), Y ~ exp(λ = 1) and Z ~ exp ( λ = 3)

Find P(max(X,Y) > 10) and E(max(X,Z)).

I know that for exponential distribution

max(X,Y) = min(X,Y) + R. (Where R is the remaining lifetime).

However I don't know how to use this fact to solve a problem

Answer 1

\begin{align} & \frac d {dx} \Pr( \max\{X,Z\} \le x) = \frac d {dx} \Pr(X\le x\ \&\ Z\le x) \\[10pt] = {} & \frac d {dx} \big( (1 - e^{-x})(1-e^{-3x}) \big) = \frac d{dx}( 1 -e^{-x} - e^{-3x} + e^{-4x}) \\[8pt] = {} & e^{-x} + 3e^{-3x} - 4e^{-4x}. \end{align} So look for $\displaystyle\int_0^\infty x(e^{-x} + 3e^{-3x} - 4e^{-4x}) \, dx,$ remembering that $\displaystyle\int_0^\infty xe^{cx} \, dx = \frac 1 {c^2}. $