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STA3007_0506_t09R

STA3007_0506_t09R - STA 3007 Applied Probability Tutorial 9...

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STA 3007 Applied Probability 2005 Tutorial 9 1. Poisson Process (a) Distribution Associated with the Poisson Process i. Example 1 Let X 1 ( t ) and X 2 ( t ) be independent Poisson processes having parameters λ 1 and λ 2 , respectively. What is the probability that X 1 ( t ) = 1 before X 2 ( t ) = 1 ? (b) The Uniform Distribution and Poisoon Processes i. Theorem 3.6 Let W 1 , W 2 , . . . be the occurrence times in a Poisson process of rate λ > 0. Conditioned on X ( t ) = n , the random variables W 1 , W 2 , . . . , W n have the joint probability density function f W 1 ,...,W n | X ( t )= n ( w 1 , . . . , w n ) = n ! t - n for 0 < w 1 < · · · < w n t. ii. Example 1 (P.301-P.303) Viewing a fixed mass of a certain radioactive material, suppose that alpha particles appear in time according to a Poisson process of intensity λ . Each par- ticle exists for a random duration and is then annihilated. Suppose that the successive lifetimes Y 1 , Y 2 , . . . of distinct particles are independent random variables having the common distribution function G ( y ) = Pr { Y k y } . Let M ( t ) count the number of alpha particles existing at time t . What is the probability distribution of M ( t ) under the condition that M (0) = 0. 1
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Let X ( t ) be the number of particles created up to time
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