The “random” parts of the algorithm in Self-Test Problem 1 can be written in terms of the generated values of a sequence of independent uniform (0, 1) random variables, known as random numbers. With [x] defined as the largest integer less than or equal to x, the first step can be written as follows:
Step 1. Generate a uniform (0,1) random variable U. Let X = [mU] + 1, and determine the value of n(X).
(a) Explain why the above is equivalent to step 1 of Problem 1,
(b) Write the remaining steps of the algorithm in a similar style.
Problem 1
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 λ1, λ2, and λ3. Let Xi denote the time at which component i fails, i = 1,2. The random variables X1,X2 are said to have a joint bivariate exponential distribution. Find P{X1 > s, X2 > t}.
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