adding two different distributions example:Gaussian and Poisson distribution

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Anusha S S
Anusha S S 2016-6-6
评论： Torsten 2016-6-8
if we add two different distributions namely gaussian which as mean and standard deviation as variables and Poisson distribution with lambda variable how to mathematically relate the resultant distribution(What distribution the resulting value will take) and how to code it

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Torsten
Torsten 2016-6-6
编辑：Torsten 2016-6-6
If X ~ Poisson(lambda), Y ~ N(mu,sigma^2), X, Y independent and Z=X+Y, then the cdf of Z is given by
P(Z<=z) = sum_{k=0}^{k=oo} P(X=k) * P(Y<=z-k).
P(X=k) = lambda^k/k! * exp(-lambda)
P(Y<=z-k) = 0.5*(1+erf((z-k-mu)/sqrt(2*sigma^2))) (erf: error function)
If needed, you can get the pdf of Z by differentiating the sum with respect to z.
Best wishes
Torsten.
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Torsten
Torsten 2016-6-8
So to get the cfd F_Z of Z=X+Y, you have to evaluate the infinite sum
F_Z(z)= sum_{k=0}^{k=Inf} lambda^k/k!*exp(-lambda)*0.5*(1+erf((z-k-mu)/sqrt(2*sigma^2)))
for different values of z.
Make an attempt. If it does not work, post the code with the error message you get.
Best wishes
Torsten.

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