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.
On
Hint: You could apply Wald's identities:
Let $(B_t,\mathcal{F}_t)_t$ be a BM$^1$ and $\tau$ an $\mathcal{F}_t$ stopping time such that $\mathbb{E}\tau<\infty$. Then $B_\tau \in L^2$ and $$\mathbb{E}B_\tau = 0 \qquad \qquad \mathbb{E}B_{\tau}^2 = \mathbb{E}\tau$$
In this case you can define $\mathcal{F}_t$ as $\mathcal{F}_t := \sigma\{(B_s, s \leq t),T_1\}$ (which is an admissible filtration since $T_1$ is independent of the BM) and use $\mathbb{E}T_1 = \frac{1}{\lambda}$.
I am going out on a limb here but maybe try the following:
We know that $$E(X) = E[E(X|T)]$$$$\text{Var}(X) = E(\text{Var}(X|T))+\text{Var}(E(X|T))$$ from the law of total variance.