CS70 Guide
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# Conditional Expectation and Variance

Properties:

### Conditional Expectation

$E(X|Y)$
is the conditional expectation of
$X$
given
$Y$
• $E(X|Y=y)$
is a fixed value, but
$E(X|Y)$
is a random variable (it is a function of
$Y$
)
• Iterated expectation:
$E(E(X|Y)) = E(X)$
$E(Y+Z | X) = E(Y|X) + E(Z|X)$
• does not work on the right hand side:
$E(Y | X+Z) \ne E(Y|X) + E(Y|Z)$
• Linearity:
$E(aX + b | Y) = aE(X|Y) + b$
• Conditioning on the same variable:
$E(g(S)T | S) = g(S)E(T|S)$

### Conditional Variance

If
$Var(Y)$
is difficult to find directly, we can use the variance decomposition to condition the variance on another variable.
$Var(Y) = E(Var(Y|X)) + Var(E(Y|X))$