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The joint probability distribution of $X_1$and $X_2$.

There are two components and three types of shocks.

Component 1 fails in a type 1 shock, component 2 fails in a type 2 shock, and components 1 and 2 fail in a type 3 shock.

Times are exponential random variables that are independent of one another.

With the corresponding rates,

${位}_1,{位}_2,\text{and}{位}_3\text{.}$

$X_i:$time at which component i fails, i = 1. 2

A bivariate exponential distribution is shared by$X1andX2$

Let $T_1,T_{2}and{T}_3$be random variables that represent the moments when type 1 shock, type 2 shock, and type 3 shock occur, respectively.

as a result

$\begin{array}{r}{T}_i~\mathrm{Expo}鈦�\left({位}_i\right)\text{Now,}\\ \text{For}{X}_1>t,X_2>s\end{array}$

Type 1 shocks take longer than t, type 2 shocks take longer than s, and type 3 shocks take longer than the maximum of t and s

Therefore

$\begin{array}{c}P\left(X_1>t,X_2>s\right)=P\left(T_1>t,T_2>s,T_3>max(s,t)\right)\\ =P\left(T_1>t\right)P\left(T_2>s\right)P\left(T_3>max(s,t)\right)\\ =e^{鈭�{位}_{1}t}鈰�e^{鈭�{位}_{2}s}鈰�e^{鈭�{位}_{3}max(s,t)}\\ =e^{鈭�\left({位}_{1}t+{位}_{2}s+{位}_{3}max(s,t)\right)}\end{array}$

Therefore$P\left(X_1>t,X_2>s\right)=e^{-\left({位}_{1}t+{位}_{2}s+{位}_3\max (s,t)\right)}$