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Deriving probability distributions

  1. Mar 8, 2007 #1
    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??
     
  2. jcsd
  3. Mar 8, 2007 #2
    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.
     
    Last edited: Mar 8, 2007
  4. Mar 10, 2007 #3
    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
    means "the random variable X given alpha"?
    But what exactly, (in words), do you mean by

    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. Mar 10, 2007 #4
  6. Apr 18, 2007 #5
    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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