A Conditional Probability Mass Function

probability bubble

conditioning

Y given X

X given Y

x	1	2	3	4	5
P(X = x)	0.092	0.210	0.395	0.210	0.092
E[Y \| X = x]	3.000	3.000	3.000	3.000	3.000

E[Y] = 3.000

The probability mass function of a pair of discrete random variables

(X,Y)

is the function

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

. The conditional mass function of

Y

given

X

is the function

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

. Thus the mass function (left-hand plot) computes probabilities of intersections, while the conditional mass function (right-hand plot) computes conditional probabilities. For each

x

value, the slice through the conditional mass function at that value gives the distribution of

Y

when

X

assumes the value

x

. The mean of this distribution is the conditional expectation of

Y

given

X=x

,

E[Y|X=x]

. The weighted average of the conditional expectations, with the weights given by the probability that

X=x

,

is the expected value of

Y

.

You can reverse the roles of

X

and

Y

to obtain the conditional mass function of

X

given

Y

and the conditional expectation of

X

given

Y=y

.