# Marginal Density

When we are considering a bivariate random variable (X, Y), the probability pertaining to one of the variables, such as P(x < X ^ x2) or P(yj < У < yf), is called the marginal probability. The following relationship between a marginal probability and a joint probability is obviously true.

figure 3.5 Domain of a double integral for Example 3.4.4

(3.4.18) P(x1 < X < x2) = P{xx < X < x2, -«з < Y < <*>).

More generally, one may replace x ^ X < x2 in both sides of (3.4.18) by x Є S where 5 is an arbitrary subset of the real line.

Similarly, when we are considering a bivariate random variable (X, Y), the density function of one of the variables is called the marginal density. Theorem 3.4.1 shows how a marginal density is related to a joint density.

THEOREM 3.4.1 Let f(x, y) be the joint density of X and Y and let f{x) be the marginal density of X. Then

(3.4.19) f(x) =

Proof. We only need to show that the right-hand side of (3.4.19) satisfies equation (3.3.1). We have

fix, y)dy dx = РІХ < X ^ xg, —00 <Y< °°) by (3.4.1)

= P(xx<X<x2) by (3.4.18). □

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