adding two different distributions example:Gaussian and Poisson distribution

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

[Image Analyst]

What does "relate" mean to you? The new distribution will be the sum of the two you summed. What else do you need to know?

Answer 1

[Torsten]

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.

[Anusha S S]

Sir,

Thanks for reply But i am unable to understand this as i am a beginner in this,can you please brief me on this ....May i know how to code this. . .

[Torsten]

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.

[Torsten]

In Mathematica, the result for lambda=0.5, mu=0 and sigma=1 can be seen here:

http://www.wolframalpha.com/input/?i=Plot+%5BTable%5BSum+%5B0.5%5Ek%2Ffactorial(k)*Exp%5B-0.5%5D*0.5*(1%2BErf%5B(x-k)%2FSqrt%5B2%5D%5D),%7Bk,0,n%7D%5D,%7Bn,20,20%7D%5D,%7Bx,-10,10%7D%5D

Best wishes

Torsten.