Suppose Antarctica has $N \sim \mathrm{Poisson}(\lambda)$ penguins. Each penguin is independently cute with probability $p$ (and not cute with probability $1-p$). Let $X$ and $Y$ be the numbers of cute and not cute penguins. Let $D = X - Y$. Find $E(D)$ and $\mathrm{Variance}(D)$.
For this problem, I thought that $D$ could be modeled by $\sim \mathrm{Poisson}(\lambda p - \lambda (1 - p))$ or $\sim \mathrm{Poisson}(\lambda (2p - 1))$. That would male $E(D) = \lambda (2p - 1)$.
However, apparently, the variance of $D$ is $\lambda$, not $\lambda (2p - 1)$ as one would expect from a Poisson random variable. Why is that?