Using discrete and continuous rvs in a distribution

Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters

Suppose also that given that , the number of accidents during that day - call it N - has a poisson distribution with mean .

Show that the conditional distribution of W given that N = n is the gamma distribution with parameters

I would like some help to write the formula for the second supposition, as it goes from a continuous rv to a discrete one

Thanks

Re: Using discrete and continuous rvs in a distribution

So I'm working on this same problem and wondering, how is the claim of proportionality that you use here justified? I thought that it would be

so that when you divide, you're not dividing by a constant but instead dividing by a function of .

Re: Using discrete and continuous rvs in a distribution

Yeah, that's fine. For finding the law of W|X you can think of all the stuff on the right side of the conditioning bar as being constants when you do any proportionality stuff.

Re: Using discrete and continuous rvs in a distribution

Oh right, duh, the thing we are to prove is that the resulting distribution has parameters which are themselves functions of ! Making sense now, thank you!

Re: Using discrete and continuous rvs in a distribution

Okay, I lied, I've been off-and-on staring at this some more and I'm back to not really getting it. I intuitively understand the idea of how, conditional on , things in terms of are like a constant, but I'm not sure how to make rigorous use of that idea.

Here's an outline of what I've done, followed by a more detailed description if it's helpful.

By some simple algebraic manipulation and Bayes's Law, I get

Where the expressions in the numerator are described in the assumptions of the problem. From that, I combine expressions with a base of and with a base of . The result is , which seems to me the (as you call it) "kernel" of a gamma distribution with parameters . Now I know that you earlier said this cannot be right, and maybe that's why I'm running into problems--however, I'm not seeing how what I've done is wrong or how anything else could work.

But as a result of so organizing my terms, my "coefficient" is now

For this to truly be a gamma distribution in those parameters, I need my coefficient to be

So how do I do this? I don't have freedom to choose what any of the terms are, so it doesn't seem like I am able to compensate for this difference by assigning some value to a constant coefficient or anything like that.

*A more detailed derivation of the expression that I ultimately obtain:*

Re: Using discrete and continuous rvs in a distribution

By the way, I just noticed that, in your earlier statement you were using the Poisson distribution, which is what the original poster had originally posted.

However, in the original poster's reply (time-stamped May 18th 8:51 AM) when he wrote exactly what the problem was asking, he wrote the exponential distribution. So really, this problem should be about each being exponential with rate .