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$N(t)$ is a Poisson process with parameter $\lambda> 0$, and $X_1,X_2,...$ are independent and identically distributed random variables with a common mean and positive variance. Let $$L(t)=\sum_{i=1}^{N(t)}X_i.$$ Find $E[L(t)|N(t) = n]$.

Any help with this is appreciated.

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Please learn LaTeX, or rather, its small subset allowing to post maths here. –  Did Oct 11 '13 at 6:08

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Hint: $$ {\rm E}[L(t)\mid N(t)=n]={\rm E}[L(t)\mathbf{1}_{\{N(t)=n\}}]/P(N(t)=n), $$ where $1_A$ denotes the indicator function of the set $A$.

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