For $\mathbb{E}[X_1|X_2]$ I am tempted to say the answer is $X_1$ because knowing an event that happened after another one does not effect the prior event's distribution. However, I am not sure if this reasoning is valid. Additionally, I am completely lost for the second part $\mathbb{E}[X_2|X_1]$.
2026-03-31 07:57:21.1774943841
Say ${\{X_t\}}_{ t \geq 0}$ is a Poisson process with rate parameter $\mu$. What are $\mathbb{E}[X_1|X_2]$ and $\mathbb{E}[X_2|X_1]$?
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Hint: $X_2 = X_1 + Y$ where $Y$ is Poisson with parameter ... and independent of ...