INTRODUCTION TO STATISTICS AND ECONOMETRICS

Conditional Density

We shall extend the notion of conditional density in Definitions 3.3.2 and

3.3.3 to the case of bivariate random variables. We shall consider first the situation where the conditioning event has a positive probability and second the situation where the conditioning event has zero probability. Under the first situation we shall define both the joint conditional density and the conditional density involving only one of the variables. A gener­alization of Definition 3.3.3 is straightforward: definition 3.4.2 Let (X, У) have the joint density fix, y) and let S be a subset of the plane such that P[(X, Y) Є 5] > 0. Then the conditional density of (X, Y) given (X, Y) Є S, denoted by f(x, у | S), is defined by

(3.4.21) f(x, у I S) =------------------------ for (x, у) Є S,

P[(X, F)ES]

= 0 otherwise.

We are also interested in defining the conditional density for one of the variables above, say, X, given a conditioning event involving both X and Y. Formally, it can be obtained by integrating f(x, у S) with respect to y. We shall explicidy define it for the case that S has the form of Figure 3.3.

DEFINITION 3.4.3 Let (X, Y) have the joint density f(x, y) and let S be a subset of the plane which has a shape as in Figure 3.3.We assume that P[(X, F) Є S] > 0. Then the conditional density of X given (X, Y) Є S, denoted by /(x I 5), is defined by

(3.4.22)

f(x I S)

otherwise.

For an application of this definition, see Example 3.4.8.

It may be instructive to write down the formula (3.4.22) explicidy for a simple case where a = — oo, b = °o, h(x) = yi, and g(x) — y2 in Figure 3.3. Since in this case the subset S can be characterized as уг ^ Y ^ y2, we have

The reasonableness of Definition 3.4.3 can be verified by noting that when

(3.4.23) is integrated over an arbitrary interval [xj, x2], it yields the con­ditional probability P(xі < X < x21 yi ^ Y — y2).

Next we shall seek to define the conditional probability when the con­ditioning event has zero probability. We shall confine our attention to the case where the conditioning event S represents a line on the (x, y) plane: that is to say, S = {(x, у) у = yi + ex], where y and c are arbitrary constants.

We begin with the definition of the conditional probability P(x < X ^ x2 I Y = yi + cX) and then seek to obtain the function of x that yields this probability when it is integrated over the interval [x1; x2], Note that this conditional probability cannot be subjected to Theorem 2.4.1, since P(Y = yi + cX) = 0.

definition 3.4.4 The conditional probability that X falls into [xb x2] given Y = yi + cX is defined by

(3.4.24) P{x — X ^ x21 Y = y + cX)

= lim P(xі ^ X ^ x21 yi + cX ^ Y ^ y2 + cX),

У2~^Уі

where yi < y2.

Next we have definition 3.4.5 The conditional density of X given Y = уi + cX, denoted by /(x I T = yi + cX), if it exists, is defined to be a function that satisfies

(3.4.25) P(xi < X < x2 I Y = yj + cX) for all x1; x2 satisfying x ^ x2.

Now we can prove

THEOREM 3.4.2 The conditional density f(xY = yi + cX) exists and is given by

f{x, yi + cx)

*00

f(x, yi + cx)dx

J —CO provided the denominator is positive.

Proof We have

(3.4.27) lim Pixi ^ X ^ x2 y + cX ^ Y ^ y2 + cX)

by the mean value theorem of integration.

Therefore the theorem follows from (3.4.24), (3.4.25), and (3.4.27). □

For an application of Theorem 3.4.2, see Example 3.4.9. An alternative way to derive the conditional density (3.4.26) is as follows. By putting a = — oo, b = oo, hix) = yi + cx, and g(x) = y2 + cx in Figure 3.3, we have from (3.4.22)

Then the formula (3.4.26) can be obtained by defining (3.4.29) fix I Y = y + cX) = lim fix yi + cX ^ Y ^ y2 + cX).

У2~>Уі

A special case of Theorem 3.4.2 where c = 0 is important enough to write as a separate theorem:

theorem 3.4.3 The conditional density of X given Y = y, denoted by f(x I yi), is given by

(3.4.30) f(x I yi) = .

ДУ l)

provided that f{y) > 0.

Figure 3.6 describes the joint density and the marginal density appear­ing in the right-hand side of (3.4.30). The area of the shaded region represents fiyf).