Let $X$ and $Y$ be jointly normally distributed $(X,Y) \in \mathcal{N}(\mu, \Sigma)$. Is the following always correct $$ P(X \leq a, X + Y \leq b) = E(P(X \leq a, X + Y \leq b|X)) $$
2026-04-12 13:29:23.1776000563
relation of probability and conditional expectation
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Yes, here are the details $$\begin{align} \mathbb{P}(X \leq a, X + Y \leq b) &=\mathbb{E}(\mathbf{1}_{\{X \leq a, X + Y \leq b \}}) \\ &=\mathbb{E}(\mathbb{E}(\mathbf{1}_{\{X \leq a, X + Y \leq b \}}|X)) \\ &= \mathbb{E}(\mathbb{P}(X \leq a, X + Y \leq b|X)) \end{align}$$ And the statement holds true for all random variables $X,Y$ (not necessarily follows joint normal distribution).