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
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This note was uploaded on 05/21/2011 for the course STA 3007 taught by Professor Kb during the Spring '11 term at CUHK.

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STA3007_0506_t09R - STA 3007 Applied Probability Tutorial 9...

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