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Unformatted text preview: STA3007 Applied probability (0809) Solution to Assignment 10 Chapter 6 Problem 2.1 Let W i be the waiting time of the i th death, and W = 0, S i be the sojourn time in the i th state, we have Pr( X ( T ) = 0) = Pr( T > W N ) = N Y i =1 Pr( T > W i  T > W i 1 ) = N Y i =1 Pr( T > W i 1 + S i  W i 1 ) = N Y i =1 Pr( T > S i ) = N Y i =1 Z ∞ Z ∞ s θ exp( θt ) dt μ i exp( μ i s ) ds = N Y i =1 μ i θ + μ i by using the memoryless property of exponential distribution. Problem 2.2 Method 1: Invoking how to get the differential equations for the pure birth process, we can get such similar differential equations for the pure death process as follows P N ( t ) = μ N P N ( t ) = θP N ( t ) , P n ( t ) = μ n P n ( t ) + μ n +1 P n +1 ( t ) = θP n ( t ) + θP n +1 ( t ) , < n < N, P ( t ) = μ 1 P 1 ( t ) = θP 1 ( t ) , with initial conditions P N (0) = 1 , P n (0) = 0 , n < N....
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 Spring '11
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