Conditional Expectation and Conditional Variance

Let f(x, y) be the joint PDF of random variables X and Y. The conditional expectation of X, given Y = y, is defined as

xf(x | Y = y) dx if X is continuous where E(X | Y = y) means the conditional expectation of X given Y = y and where f (x | Y = y) is the conditional PDF of X. The conditional expectation of Y, E(Y | X = x), is defined similarly.

Conditional Expectation. Note that E(X | Y) is a random variable because it is a function of the conditioning variable Y. However, E(X | Y = y), where y is a specific value of Y, is a constant.

Conditional Variance. The conditional variance of X given Y = y is defined as var (X | Y = y) = E{[X - E(X | Y = y)]2 | Y = y}

= Y,[X - E(X | Y = y)]2 f (x | Y = y) if X is discrete

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