I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. How do I calculate the expected value of $X$ given this information? I.e. I want to calculate $$ E[X∣Z₁=c₁∧Z₂=c₂]. $$
2026-03-27 11:48:05.1774612085
Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables
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