Page 54 - Elements of Distribution Theory

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052184472Xc02 CUNY148/Severini May 24, 2005 2:29

40 Conditional Distributions and Expectation

Let F X denote the distribution function of the marginal distribution of X. Clearly, F X
and F are related. For instance, for any A ⊂ X,

Pr(X ∈ A) = dF X (x).
A
Hence, F X must satisfy

dF X (x) = dF(x, y)
A A×Y
for all A ⊂ X.
The cases in which (X, Y) has either an absolutely continuous or a discrete distribution
are particularly easy to handle.

Lemma 2.1. Consider a random vector (X, Y), where X is d-dimensional and Y is q-
dimensional and let X × Y denote the range of (X, Y).
(i) Let F(x 1 ,..., x d , y 1 ,..., y q ) denote the distribution function of (X, Y). Then X
has distribution function
F X (x 1 ,..., x d ) = F(x 1 ,..., x d , ∞,..., ∞)
≡ lim ··· lim F(x 1 ,..., x d , y 1 ,..., y q ).
y 1 →∞ y q →∞
(ii) If (X, Y) has an absolutely continuous distribution with density function p(x, y)
then the marginal distribution of X is absolutely continuous with density function

p X (x) = p(x, y)dy, x ∈ R d
R q
(iii) If (X, Y) has a discrete distribution with frequency function p(x, y), then the
marginal distribution of X is discrete with frequency function of X given by

p X (x) = p(x, y), x ∈ X.
y∈Y
Proof. Part (i) follows from the fact that

Pr(X 1 ≤ x 1 ,..., X d ≤ x d ) = Pr(X 1 ≤ x 1 ,..., X d ≤ x d , Y 1 < ∞,..., Y q < ∞).

Let A be a subset of the range of X. Then, by Fubini’s Theorem (see Appendix 1),

Pr(X ∈ A) = p(x, y) dy dx ≡ p X (x) dx,
A R q A
proving part (ii). Part (iii) follows in a similar manner.

Example 2.1 (Bivariate distribution). Suppose that X and Y are both real-valued and that
(X, Y) has an absolutely continuous distribution with density function

p(x, y) = 6(1 − x − y), x > 0, y > 0, x + y < 1.