# What is distribution of $Z = X + Y$ where $X$ is Poisson distributed and $Y$ is normally distributed?

How does one calculate the distribution of a random variable $Z = X + Y$, where $X$ is Poisson distributed and $Y$ is normally distributed and independent of $X$? How to do that in general case? Do you know any textbook about this topic? It's really hard for me to seach on this topic, because I don't know what to call this problem.

• See any probability textbook. The distribution function of sum of iid variables is convolution of individual distribution functions. Apr 6, 2012 at 8:06
• @mpiktas But X and Y are not identically distributed: X is poissonian and Y is gaussian. Apr 6, 2012 at 19:08
• @tomas, I meant to write id. Independence is the key here. Apr 7, 2012 at 8:26

Here I would calculate the distribution of $Z=X+Y$ using conditional decomposition. If $X \sim {\cal P}(\lambda)$ and $Y \sim {\cal N}(\mu, \sigma^2)$ then the conditional distribution of $Z$ given $X=x$ is the distribution of $x+Y$ which is the normal distribution ${\cal N}(\mu+x, \sigma^2)$. Finally, denoting by $f_{\mu, \sigma}$ the density function of ${\cal N}(\mu, \sigma^2)$, the density of $Z$ is the function $$z \mapsto \sum_{x \geq 0}p_x f_{\mu+x, \sigma}(z)$$ where $p_x=\Pr(X=x)=\exp(-\lambda)\frac{\lambda^x}{x!}$. Of course it remains to do the calculation of this sum.