$X^2_0 \sim Poisson(\lambda T)$, $X^1_0 \sim Poisson(\lambda T)$, $X^2_1 \sim Poisson(\lambda(T+\alpha X^2_0 -\alpha X^1_0))$ with $\alpha, T, \lambda$ being constants. $X^2_0, X^1_0$ are independent of each other.
I want to find E($X^2_1$). I have two choices, but I am not sure which one is correct?
The first choice is: since $X^2_0$ and $X^1_0$ have the same distribution, $X^2_0 \sim Poisson(\lambda T)$, so $E(X^2_1)=\lambda T$.
The second choice is $E(X^2_1)=E(E(X^2_1 \mid X^2_0=x, X^1_0=y))$