conditional distribution1

CVondintional Distribution

One of the most useful concepts in probability theory is that of conditional probability

and conditional expectation. The reason is twofold. First, in practice, we

are often interested in calculating probabilities and expectations when some partial

information is available; hence, the desired probabilities and expectations are

conditional ones. Secondly, in calculating a desired probability or expectation it is

often extremely useful to first “condition” on some appropriate random variable.

The Discrete Case

Recall that for any two events E and F, the conditional probability of E given F

is defined, as long as P(F)>0, by

P(E|F) = P(EF)/P(F)

Hence, if X and Y are discrete random variables, then it is natural to define the

conditional probability mass function of X given that Y = y, by

pX|Y (x|y) = P{X = x|Y = y}

= P{X = x,Y = y}/P{Y = y}

= p(x, y)/pY (y)


for all values of y such that P{Y = y} > 0. Similarly, the conditional probability

distribution function of X given that Y = y is defined, for all y such that

P{Y = y} > 0, by

FX|Y (x|y) = P{X _ x|Y = y}

=?pX|Y (a|y)

Finally, the conditional expectation of X given that Y = y is defined by

E[X|Y = y] =?xP{X = x|Y = y}

In other words, the definitions are exactly as before with the exception that
everything is now conditional on the event that Y = y. If X is independent of Y ,
then the conditional mass function, distribution, and expectation are the same as
the unconditional ones. This follows, since if X is independent of Y , then
pX|Y (x|y) = P{X = x|Y = y}= P{X = x}
Example 3.1 Suppose that p(x, y), the joint probability mass function of X
and Y , is given by
p(1, 1) = 0.5, p(1, 2) = 0.1, p(2, 1) = 0.1, p(2, 2) = 0.3
Calculate the conditional probability mass function of X given that Y = 1.