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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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    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 $0$.2012-05-18

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