Joint Bivariate Exponential Distribution

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Homework Statement
Consider two components and three types of shocks. A type 1 shock causes component 1 to fail, a type 2 shock causes component 2 to fail, and a type 3 shock causes both components 1 and 2 to fail. The times until shocks 1, 2, and 3 occur are independent exponential random variables with respective rates [itex]\lambda_1, \lambda_2, \lambda_3[/itex]. Let [itex]X_i[/itex] denote the time at which component i fails, i = 1, 2. The random variables [itex]X_1, X_2[/itex] are said to have a joint bivariate exponential distribution. Find [itex]P\{X_1 > s, X_2 > t\}[/itex].

The attempt at a solution
This problem would by so much easier if type 3 shocks didn't exists as it would make [itex]X_1, X_2[/itex] independent. Anywho...

Let [itex]Y_1, Y_2, Y_3[/itex] be the times shocks of type 1, 2, 3 occurred. I know I'm going to have to deal with the joint distribution of these three random variables. However, I can't think of anything. I need a little hint.
 

Answers and Replies

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One idea is suggested in Yan, Carpenter and Diawara (2006) and in Diawara and Carpenter (2008) in their papers from AJMMS. Please check it out, and let me know if you have questions.
 

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