Question

Derivation of conditional probability density function

Answer 1

Suppose X and Y are continuous random variables with joint probability density function f(x,y) and marginal probability density functions fX(x) and fY(y), respectively. Then, the conditional probability density function of Y given X = x is defined as:

h(y|x)=f(x,y)fX(x)

provided fX(x) > 0. The conditional mean of Y given X = x is defined as:

E(Y|x)=∫∞−∞yh(y|x)dy

The conditional variance of Y given X = x is defined as:

Var(Y|x)=E{[Y−E(Y|x)]2|x}=∫∞−∞[y−E(Y|x)]2h(y|x)dy

or, alternatively, using the usual shortcut:

Var(Y|x)=E[Y2|x]−[E(Y|x)]2=[∫∞−∞y2h(y|x)dy]−μ2Y|x