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.