Poisson Distributions with Conditional Probability

Date: 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/