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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.
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.