Some components of a two-component system fail after receiving a shock.

Shocks of three types arrive independently and in accordance with Poisson processes. Shocks of the first type arrive at a Poisson rate λ1 and cause the first component to fail. Those of the second type arrive at a Poisson rate λ2 and cause the second component to fail. The third type of shock arrives at a Poisson rate λ3 and causes both components to fail. Let X1 and X2 denote the survival times for the two components. Show that the joint distribution of X1 and X2 is given by

This distribution is known as the bivariate exponential distribution.

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P{X1>s, X>t} = exp{-1s-2t 3 max(s, t)}