# A Conditional Probability Mass Function

A Conditional Probability Mass Function

The probability mass function of a pair of discrete random variables is the function . The conditional mass function of given is the function . Thus the mass function (left-hand plot) computes probabilities of intersections, while the conditional mass function (right-hand plot) computes conditional probabilities. For each value, the slice through the conditional mass function at that value gives the distribution of when assumes the value . The mean of this distribution is the conditional expectation of given , . The weighted average of the conditional expectations, with the weights given by the probability that , is the expected value of .

(X,Y)

f(x,y)=P(X=x,Y=y)

Y

X

f(y|x)=P(Y=y|X=x)

x

Y

X

x

Y

X=x

E[Y|X=x]

X=x

Y

You can reverse the roles of and to obtain the conditional mass function of given and the conditional expectation of given .

X

Y

X

Y

X

Y=y