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