Distribution of sum of normal and gamma random variable

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November 27th 2013, 07:10 AM
usagi_killer

Distribution of sum of normal and gamma random variable

Let $Z = X + Y$ where $X \sim N\left(\mu, \sigma^2 \right)$ and $Y \sim \Gamma\left(k, \theta \right)$ using this parametrization of the Gamma distribution. Also assume $X$ and $Y$ are independent. Then what is the distribution (pdf) of $Z$ ?

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This question doesn't seem as straightforward as it sounds. For example, I have tried using the convolution formulas here, but can't seem to find a closed form expression for the integral. I have also tried multiplying the moment generating functions (mgfs) of $X$ and $Y$ , but it does not seem to match up to any known mgfs.

Does anyone have any ideas on how to find the distribution for $Z$ ?
November 27th 2013, 05:58 PM
romsek

Re: Distribution of sum of normal and gamma random variable

I doubt that this will get you a closed form but did you try using characteristic functions?
November 27th 2013, 10:17 PM
usagi_killer

Re: Distribution of sum of normal and gamma random variable

Quote:

Originally Posted by romsek

I doubt that this will get you a closed form but did you try using characteristic functions?

I'm not too sure what characteristic functions are, how would I apply that here?
November 27th 2013, 10:31 PM
romsek

Re: Distribution of sum of normal and gamma random variable

it's basically the fourier transform of a probability distribution function and as such the characteristic function of the sum of 2 rvs is the product of the individual characteristic functions.

I just came across this

http://cran.r-project.org/web/packag...ormalGamma.pdf

it won't give you a closed form answer but it might be of interest.