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Is the probability density function (pdf) of the Compound Poisson $X(t)=\sum_{i=1}^{N(t)}Y$ known? Where $N(t)$ is a Poisson process and $Y$ is normally distributed with mean $\mu$ and variance $\sigma^2$. I know its moment generating function (mgf).

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It doesn't have a pdf. There is a positive probability that $N(t)=0$ and so $P(X(t)=0) > 0$; $X$ is not a continuous random variable. Were you intending to condition on $N(t) > 0$ or something? – Nate Eldredge May 18 '12 at 13:59
yes, note Nates comment, but conditionally on $N(t) = j$ it is a $N(j \mu, j \sigma^2)$ density and you sum them up. $j=0$ gives you that delta function at $0$. – mike May 18 '12 at 14:26

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