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let $T_1$ be the first occurrence of a Poisson process at rate $\lambda$, and $X(t) = \sigma B(t) + \mu t$ be another independent Brownian motion with drift, calculate $E(X(T_1))$ and $\operatorname{Var}(X(T_1))$.

I know $E(T_1) = 1/\lambda$, as well as the following

$E(X(t))=\mu t$
$\operatorname{Var}(X(t)) = \sigma^2 t$

but not sure what's the result when $t$ becomes a random variable itself.

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    Is $T_1$ supposed to be independent of the Brownian motion $B$? Try conditioning on $T_1$.2012-12-12
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    Yes, $T_1$ is independent of the Brownian motion $B$. Would you mind explain a bit more how shall I condition on $T_1$?2012-12-12

2 Answers 2