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
That shouldn't be what you get when you evaluate the RHS. Come to think of it, you didn't even define what is...I think what you intended was that is distributed Poisson with mean W, and you want the distribution of .
Can you post the question exactly as it is written? Among other things that don't make sense, if you interpret the question as I wrote it above you get and not .
When I say the "kernal" I mean the part of the density that matters, i.e. everything but a normalizing constant (in this case, the x_i are considered fixed so you can get rid of anything that is only a function of the x_i as well as any other fixed constants). You can retrieve the normalizing constant because the density must integrate to 1. If you can show that a pdf is proportional to the kernal of something you know then you know what the pdf is because you can get the normalizing constant by integrating the kernal.
The technique I used is nice because it saves you from having to calculate the marginal of X which requires integration.
To give another example, here are a couple of useful kernals for the normal distribution: as well as .
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 .
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:
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 .