1. ## Jump-diffusion mean

I have a problem calculating the mean for the following probability distribution,

exp(sum(X))

where the sum is Poisson-distributed with intensity lamda and X is a normal distribution with mean my and variance V, i.e. the number of normal variables is Poisson distributed.

2. Originally Posted by McGoose
I have a problem calculating the mean for the following probability distribution,

exp(sum(X))

where the sum is Poisson-distributed with intensity lamda and X is a normal distribution with mean my and variance V, i.e. the number of normal variables is Poisson distributed.
Let's take $\displaystyle \mu = 0, \, \sigma = 1$ to keeps things simple.
Conditioned on $\displaystyle N$ = number of terms, $\displaystyle Y_N = \sum_{k=1}^N X_k$ has a normal distribution with mean 0 and standard deviation $\displaystyle \sqrt{N}$. Then $\displaystyle e^{Y_N}$ has mean $\displaystyle e^{N/2}$ (lognormal distribution with location parameter 0). Now take the expectation with respect to N:
$\displaystyle \sum_{n=0}^\infty e^{-\lambda} \frac{\lambda^n e^{n/2}}{n!} = e^{(\sqrt{e}-1)\lambda}$.