Let $Z(t)=\sum_{i=1}^{N(t)} X_i$ and let $N(t)$ be a Poisson process with parameter $\lambda$ and $X_1,X_2,\dots$ positive iid random variables with density function $f_X(x)$, independent of $N(t)$.How can I calculate $E[e^{\lambda t-\mu Z(t)}]$?
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Let $Z(t)=\sum_{i=1}^{N(t)} X_i$ and let $N(t)$ be a Poisson process with parameter $\lambda$ and $X_1,X_2,\dots$ positive iid random variables with density function $f_X(x)$, independent of $N(t)$.How can I calculate $E[e^{\lambda t-\mu Z(t)}]$?