Conditional Expectation of Random Variables.

Question

Let $X$ and $Y$ be two random variables defined in the same space. Such that $X + Y \sim \mathcal{P}(\lambda)$ with $\lambda > 0$.

Show that $ X|X + Y = k \sim Bi(k; p)$ for some $ 0 < p < 1$ if and only if, $X$ and $Y$ are independent and have distributions: $X\sim \mathcal{P}(\lambda p)$ and $Y \sim \mathcal{P}(\lambda(1-p))$

This question is supposed to be answered using conditional expectation. I am quite lost in how to get started in either way. A hint would be greatly appreciated.

Answer 1

1. $X\mid X+Y = k \sim\mathcal{Bin}(k, p)$ $$\mathsf P(X{=}x\mid X{+}Y{=}k) \;=\; \dbinom{k}{x} p^{x}(1-p)^{k-x}$$

2. $X+Y\sim\mathcal P(\lambda)$ $$\mathsf P(X{+}Y{=}k) \;=\; \dfrac{\lambda^k e^{-\lambda}}{k!}$$

3. $\mathsf P(X{=}x, Y{=}k{-}x) \;=\; \mathsf P(X{=}x\mid X{+}Y{=}k)\cdot\mathsf P(X{+}Y{=}k)$

Put it together and see what falls out.