Page 64 - Elements of Distribution Theory

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50 Conditional Distributions and Expectation

Consider Pr(X = 1|Y = 0). Since X and Y are independent, it follows from Example 2.9
that
1
Pr(X = 1|Y = 0) = Pr(X = 1) = .
2
Note that the event Y = 0is equivalent to the event Z = 0. Using (2.4),
c
Pr(X = 1|Z = 0) = .
c + 1
Thus, two different answers are obtained, even though the events Y = 0 and Z = 0 are
identical.
Some insight into this discrepency can be obtained by considering the conditioning events
|Y| < and |Z| < , for some small ,in place of the events Y = 0 and Z = 0, respectively.
Note that the events |Y| < and |Z| < are not equivalent. Suppose that c is a large number.
Then, |Z| < strongly suggests that X is 1 so that we would expect Pr(X = 1||Z| < )to
be close to 1. In fact, a formal calculation shows that, for any 0 < < 1,
c
Pr(X = 1||Z| < ) = ,
c + 1
while, by the independence of X and Y,
1
Pr(X = 1||Y| < ) = Pr(X = 1) = ,
2
results which are in agreement with the values for Pr(X = 1|Z = 0) and Pr(X = 1|Y = 0)
obtained above.

Conditional distribution functions and densities
Since Pr(X ∈ A|Y = y) defines a probability distribution for X, for each y, there exists a
distribution function F X|Y (x|y) such that

Pr(X ∈ A|Y = y) = dF X|Y (x|y);
A
the distribution function F X|Y (·|y)is called the conditional distribution function of X given
Y = y.By(2.2), F,thedistributionfunctionof(X, Y), F Y ,themarginaldistributionfunction
of Y, and F X|Y are related by
F(x, y) = F X|Y (x|y)F Y (y) for all x, y.

If the conditional distribution of X given Y = y is absolutely continuous, then

Pr(X ∈ A|Y = y) = p X|Y (x|y) dx
A
where p X|Y (·|y) denotes the conditional density of X given Y = y.If the conditional dis-
tribution of X given Y = y is discrete, then

Pr(X ∈ A|Y = y) = p X|Y (x|y)
x∈A
where p X|Y (·|y) denotes the conditional frequency function of X given Y = y.