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Conditional Expectation From the Joint Gaussian

"XX and YY are independent normals and you observe X+YX + Y. What is your best estimate of XX?" is one of the most common derivation questions in quant research interviews. It looks like a trick, but it is the whole of signal extraction in one line: you see a noisy total and want the part you care about. Interviewers ask you to derive the answer rather than quote it, so this lesson derives it three ways.

The general result

For XX and YY that are jointly normal, the conditional distribution of XX given Y=yY = y is normal, with

The conditional normal
E[X∣Y=y]=μX+Cov⁡(X,Y)Var⁡(Y)(y−μY),Var⁡(X∣Y)=Var⁡(X) (1−ρ2)E[X \mid Y = y] = \mu_X + \frac{\operatorname{Cov}(X, Y)}{\operatorname{Var}(Y)}(y - \mu_Y), \qquad \operatorname{Var}(X \mid Y) = \operatorname{Var}(X)\,(1 - \rho^2)

The mean moves linearly with what you observe, by the regression slope, and the variance shrinks by the share that the observation explains.

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