This should help, since you have the right idea.
NOW, plug in x, z-x and z into the distribution of the three DIFFERENT Poissons and it should clean up nicely.
I have the following problem.
Let X and Y be independent, with X~Poi and Y~Poi . Let Z = X + Y. Find the distribution X | Z = z.
Now I know that the sum of two independent random variables which both have a Poisson distribution, also has a Poisson distribution.
so the pmf of Z is
but Im having trouble calculating . This is where Im stuck.
I know that the conditional probability will return a binomial distribution. Can anyone please help me understand how to calculate and consequently obtain a binomial distribution?
Thank-you so much matheagle. That was so helpful, I understand conditional probability a lot better now.
Just one thing,
Ive managed to get the solution down to
But I can't work out how to get from my solution down to what it should be.
I bet its something really obvious but I cant see it.