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Distribution of exponential family

  1. Feb 1, 2016 #1
    Let's say my probability function is given by: p(y1,y2)=Γ(y1+y2+γ)/((y1+y2)!*Γ(γ)), when γ>0 is known. I suppose it is from an exponential family but I can't write in canonical form because I'm only familiar with exponential family with one variable so I'm confused now when there's to variable. Can someone help me out here.
     
  2. jcsd
  3. Feb 1, 2016 #2

    andrewkirk

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    For the distribution of a 2-vector random variable like that to be in the exponential family it must be able to be written in the factorised form

    $$p(y_1,y_2)=h(y_1,y_2)g(\gamma)\exp\left(\sum_{k=1}^s\eta_k(\gamma)T_k(y_1,y_2)\right)$$

    where ##s## is a non-negative integer and ##h,g, \eta_1,...,\eta_s, T_1,...,T_s## are all known functions.
     
  4. Feb 1, 2016 #3
    Consider the random variable ##Z=Y_1+Y_2##.

    Therefore, you have:

    ##
    \begin{eqnarray*}
    \frac{\Gamma(y_1+y_2+\gamma)}{(y_1+y_2)! \ \Gamma(\gamma)} = \frac{\Gamma(z+\gamma)}{z! \ \Gamma(\gamma)} = \frac{1}{z!} \cdot \frac{1}{\Gamma(\gamma)} \cdot \Gamma(z+\gamma) \\
    \end{eqnarray*}##

    Do you have a function ##h(z)## and a function ##g(\gamma)## now?

    Also, remember that ##a=\exp(\log(a))##! :wink:
     
  5. Feb 2, 2016 #4
    Thanks for the answers. I have edited my model a lot and I'm now looking at this model. probability function for Y=(Y1,Y2) is given by p. Can anyone bring this to canonical form so I can find expected value and variance matrixes.
    https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202016-02-02%20kl.%2007.35.26.png [Broken]
     
    Last edited by a moderator: May 7, 2017
  6. Feb 2, 2016 #5
    .... or just help me find the variance and expected value :)
     
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