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 $\displaystyle (t, \beta)$

Suppose also that given that $\displaystyle W = w $, the number of accidents during that day - call it N - has a poisson distribution with mean $\displaystyle w $.

Show that the conditional distribution of W given that N = n is the gamma distribution with parameters $\displaystyle (t+n, \beta +\sum_{i = 1}^n x_i)$

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

$\displaystyle f_{W|X}(w|x) P_{X}(x) = f_{X|W}(x|w)P_{W}(w)$

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

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 $\displaystyle x_{i}$! 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 $\displaystyle X$, things in terms of $\displaystyle X$ 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

$\displaystyle P_{W|X}P(w|x) = \frac{P_{X|W}(x|w)P_{W}(w)}{P_{X}(x)}$

Where the expressions in the numerator are described in the assumptions of the problem. From that, I combine expressions with a base of $\displaystyle w$ and with a base of $\displaystyle e$. The result is $\displaystyle w^{t+n-1}e^{-w(\beta +\sum x_{i})}$, which seems to me the (as you call it) "kernel" of a gamma distribution with parameters $\displaystyle t+n, \, \, \beta+\sum x_{i}$. 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

$\displaystyle \frac{\beta^{t}}{\Gamma (t)P_{X}(x)}$

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

$\displaystyle \frac{\Big( \beta+\sum x_{i} \Big)^{t+n}}{\Gamma(t+n)}$

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:*

$\displaystyle P(X_{1}=x_{1}, ..., X_{n}=x_{n}|W=w)P(W=w) \quad = \\\\ P(W=w|X_{1}=x_{1}, ..., X_{n}=x_{n})P(X_{1}=x_{1}, ..., X_{n}=x_{n}) \quad \Longrightarrow \\\\ P(W=w|X_{1}=x_{1}, ..., X_{n}=x_{n}) \quad = \quad \frac{P(X_{1}=x_{1}, ..., X_{n}=x_{n}|W=w)P(W=w)}{P(X_{1}=x_{1}, ..., X_{n}=x_{n})} \quad = \\\\ \frac{w^{n}e^{-w(x_{1}+...+x_{n})}\frac{\beta^{t}}{\Gamma (t)}w^{t-1}e^{-\beta w}}{P(X_{1}=x_{1}, ..., X_{n}=x_{n})} \quad = \quad \frac{\beta^{t}}{\Gamma (t) P(X_{1}=x_{1}, ..., X_{n}=x_{n})}{w^{t+n-1}e^{-w(\beta + \sum_{i=1}^{n}x_{i})}}$

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 $\displaystyle X_{i}|W$ being exponential with rate $\displaystyle w$.