Deriving probability distributions

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  • #1
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Suppose I had a random variable, X, that followed a Gamma distribution.
A Gamma distribution can be defined as [tex] \Gamma(\alpha,\beta) [/tex], where [tex]\alpha[/tex] and [tex]\beta[/tex] are the 'scale' and 'shape' parameters.
Now suppose if [tex]\alpha[/tex] was a random variable, say following a binomial distribution, how would I then represent the distribution of X.

I was thinking that since the parameter [tex]\alpha[/tex] now represents a random variable, the distribution of X, would simply be a binomial distribution multiplied by a Gamma distribution???
Would it be correct to do this??
 

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  • #2
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No. That is wrong.

[tex]X|\alpha[/tex] is distributed as [tex]\Gamma(\alpha,\beta)[/tex] and [tex]\alpha[/tex] is distributed as Bin(n,p).

Therefore, [tex]f(X=x|\alpha)f_\alpha(\alpha)=f_{(x,\alpha)}(x,\alpha)[/tex].

Now, to get the distribution of x, you just sum over all alpha. That is,

[tex]f_x(x)=\sum_{i=0}^nf(X=x|\alpha=i)f_\alpha(\alpha=i)[/tex].

I'm not sure what this distribution is as I haven't calculated it yet. I doubt it will reduce to something familiar.

However! If alpha was distributed as poisson then it becomes an interesting distribution which is a really good exercise.

If you don't understand any of this just say so.
 
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  • #3
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Thanks for the help ZioX, much appreciated!
I had a feeling I wasn't doing it right..... but I'm not too sure I fully understand what you're doing. I think I get the gist of what you're doing, but just getting a bit bogged down with the mathematical notation you're using.

So firstly I presume that
[tex]X|\alpha[/tex]
means "the random variable X given alpha"?
But what exactly, (in words), do you mean by
[tex]f(X=x|\alpha)f_\alpha(\alpha)=f_{(x,\alpha)}(x,\alpha)[/tex]

Also, I'm curious as to why you say, it would be interesting if alpha was distributed as Poisson, as this is one of the cases I will also be looking at!
Is there some standard distribution that comes out when you use a Poisson??
 
  • #5
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I'm slightly confused about the answer that's given here. I needed to find the distribution of [tex]X|\alpha[/tex], where [tex]X|\alpha[/tex] is distributed as a Gamma, [tex]\Gamma(\alpha,\beta)[/tex], and [tex]\alpha[/tex] is distributed as Bin(n,p).

The answer was to the (marginal) distribution of X, you sum over to get [tex]f_x(x)=\sum_{i=0}^nf(X=x|\alpha=i)f_\alpha(\alpha= i)[/tex]

But if X is gamma distributed, and a gamma distribution is a continous distribution, then shouldn't the above formulae be an intregal rather than a summation??
 

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