Conditional Distribution and Expectation

Suppose bivariate case

• Conditional Probability

$P(a \leq X_1 \leq b \space | \space X_2=x_2) =$ $\left\{\begin{matrix} \sum_{a \leq X_1 \leq b} p(x_1|x_2):discrete\\ \int_{a}^{b} f(x_1|x_2) dx_1:continuous \end{matrix}\right.$

• Conditional Expectation

$E(X_1|X_2=x_2) =$ $\left\{\begin{matrix} \sum_{x_1} x_1p(x_1|x_2) : discrete\\ \int_{-\infty}^{\infty} x_1f(x_1|x_2) dx_1 : continuous \end{matrix}\right.$

$E(g(X_1)|X_2=x_2) =$ $\left\{\begin{matrix} \sum_{x_1} g(x_1)p(x_1|x_2) : discrete\\ \int_{-\infty}^{\infty} g(x_1)f(x_1|x_2) dx_1 : continuous \end{matrix}\right.$

• Conditional Variance

$Var(X_1|X_2) = E[( X_1 - E(X_1|X_2) )^2 | X_2]= E(X_1^2|X_2) - E^2(X_1|X_2)$
pf)

Thereom)

Let $(X_1,X_2)$ be a random vector with $Var(X_2) < \infty$
① $E[E(X_2|X_1)] = E(X_2)$
② $Var(X_2)=E[Var(X_2|X_1)] + Var[E(X_2|X_1)]$
pf)