Poisson Distributions with Conditional ProbabilityDate: 07/22/99 at 21:26:47 From: Kathleen Subject: Statistics: Poisson distribution/conditional probability Hello, The question I'm trying to figure out is: Let X be Poisson(2) and independent of Y which is Poisson(3) and let W = X+Y. Find the distribution of X conditional on W = 10. I have figured out that the sum of two independent random variables, which have Poisson distribution, also has a Poisson distribution, specifically of Poisson(a_1+a_2) where a_1, a_2 are the parameters of the two independent random variables. Thus, I know W will have distribution Poisson(5). I also know what the probability functions look like for X being Poisson(2) and W being Poisson(5). Now I think (I could be wrong) that in order to find the distribution of X conditional on W = 10 I want to figure out P(X|W=10). Now using the conditional probability definition you have P(X|W=10) = P(X,W=10) --------- P(W=10) I can easily figure out P(W=10) but my problem is that I don't know how to figure out P(X,W=10). What is this intersection equal to? Could you help please? Thank you very much, Kathleen Date: 07/23/99 at 15:44:51 From: Doctor Anthony Subject: Re: Statistics: Poisson distribution/conditional probability P(X=1)* P(Y=9) 2*e^(-2) * 3^9/9! * e^(-3) P(X=1|W=10) = -------------- = -------------------------- P(W=10) 5^10/10! * e^(-5) 2 * 3^9/9! = ---------- 5^10/10! 2 * 10 * 3^9 = ------------ 5^10 393660 = ------- 9765625 = 0.04031 and the other probabilities can be found in the same way. The maximum value that X can take, given that W = 10, is of course 10. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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