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I am working with the distribution of the sum of two dependent random variables. In my problem, there are two unobserved events, X and Y, where X precedes Y and Y is a function of the outcome of X, but I only observe the sum of the two outcomes, i.e:

$$Z=X+Y|X$$ $$Pr(Z=x)=\sum_{z=0}^xPr(X=z)*Pr(Y=x-z|X=z)$$

It looks like a standard convolution distribution, except that Y and X are not independent. In my particular case, X and Y have a Poisson distribution, where the mean of Y is an increasing function of the value of the outcome of X. I am unaware of a closed form for the representation of the PMF of this distribution, and so need to derive it directly through the above sum. For my particular problem, I need to compute the PMF thousands of times, and am looking for ways to reduce the computational burden of doing so. As such, I was hoping someone would either

1) Know if there is a closed form representation for the PMF

2) Know of any approximations to the PMF

Any help would be greatly appreciated.

Thanks!

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    $\begingroup$ Your questions cannot be answered with the information given: you need to supply a formula expressing how $E[Y]$ depends on $X$. $\endgroup$ – whuber Jun 27 '14 at 21:53
  • $\begingroup$ There's an infinity of ways of having dependent Poissons. $\endgroup$ – Glen_b -Reinstate Monica Jun 28 '14 at 9:48
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You are on the right track, the conditional distribution $X|Y$ is a so-called mixed poisson distribution (the Poisson mean is just equal to its parameter $\lambda$).

You may find its explanation and density expression in these sources:

  • www.actuaries.org/LIBRARY/ASTIN/vol35no1/3.pdf
  • 30923.vws.magma.ca/LIBRARY/ASTIN/vol16s/s59.pdf

I did not find a specific solution where the mixing distribution is Poisson but maybe you can derive it from there.

If you then cannot derive the joint density as in your formula above, you may also try to find it by the Moment Generating Function: http://en.wikipedia.org/wiki/Moment-generating_function#Sum_of_independent_random_variables (where they assume independence though).

2) The convolution may be approximated in MATLAB conv: http://www.mathworks.ch/ch/help/matlab/ref/conv.html

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