a) Say we have two independent distribution variables $X \sim Pois(\lambda)$ and $Y \sim Pois(\mu)$. We know that the sum of the two will be $$X+Y\sim Pois(\lambda+\mu),$$ and hence the variance will be $Var(X+Y)=\lambda+\mu$.
b) We also know that in general $Var(b\cdot X) = b^2\cdot Var(X)$: $$ Var(bX) = E(b^2X^2) - E^2(bX) = b^2E(X^2) - b^2E^2(X) = b^2 Var(X) $$
However, if I take the distribution $X + X$, I have two different results for the variance:
a) $X + X \sim Pois(\lambda + \lambda) = Pois(\alpha)$, with $\alpha = 2\lambda$, and hence $$Var(X+X) = \alpha = 2\lambda$$ b) $$Var(X+X)=Var(2X)=2^2Var(X)=4\lambda$$
What am I doing wrong? According to my exercise sheet, a) is correct, but then I don't see why b) wouldn't apply. Thanks for any hints!