Exercise$3$:

Two subsystems S$1$ and S$2$ operate according to $3$ different assemblies

a: assembly in series; b: assembly in parallel ; b: assembly in "stand by$"($S$1$ works only

if S$2$ crashes and S$2$ works only if S$1$ crashes$)$

X $1$ is the running time of S $1$ and X $2$ are the running time of S $2($ X $1$ and X $2$ are

independent$).$ X $1$ and X $2$ are two exponential random variables:

$f_1(x)=\frac{1}{{}_1}{e}^{-(\frac{x}{{}_1})}$ for $x>0$;$f_2(x)=\frac{1}{{}_2}{e}^{-(\frac{x}{{}_2})}$ for $x>0$

$Y$ is the running time of the system. Compute $E(Y)($for the $3$ different assemblies$)$