Suppose that the number of customers arriving at a fast-food restaurant on a given day follows $\text{Poisson}(\lambda)$. Assume that each customer purchases a drink with probability $p$, independently from other customers. Let $X$ be the number of customers who purchase drinks. Let $Y$ be the number of customers that do not purchase drinks. Find $\text{E}[X^2Y^2]$
My attempt:
Here, the original Poisson process has been split into two Poisson processes, such that $X\text{~Poisson}(\lambda p)$ and $Y\text{~Poisson}(\lambda q)$. Also, since when a Poisson process is split, two independent Poisson processes are created, hence $X$ and $Y$ are independent.
So, $\text{E}[X^2Y^2]=\text{E}[X^2]\text{E}[Y^2]=(\lambda p+\lambda^2p^2)\cdot(\lambda q+ \lambda^2q^2)=\lambda^2pq\cdot(\lambda^2pq+2\lambda)$
but the correct answer is $\lambda^2pq(\lambda^2+\lambda+1)$. So, what did i do wrong?