Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

90 Multivariate Distributions

Table 2.1.1: Joint and Marginal Distributions for the discrete random vector

(X 1 ,X 2 ) of Example 2.1.1.

Support ofX 2

0123 pX 1 (x 1 )

(^018180028)
Support ofX (^1102828048)
(^200181828)
pX 2 (x 2 )^18383818
for allx 1 ∈ R. By Theorem 1.3.6 we can write this equation asFX 1 (x 1 )=
limx 2 ↑∞F(x 1 ,x 2 ). Thus we have a relationship between the cdfs, which we can
extend to either the pmf or pdf depending on whether (X 1 ,X 2 ) is discrete or con-
tinuous.
First consider the discrete case. LetDX 1 be the support ofX 1 .Forx 1 ∈DX 1 ,
Equation (2.1.7) is equivalent to
FX 1 (x 1 )=
∑∑
w 1 ≤x 1 ,−∞<x 2 <∞
pX 1 ,X 2 (w 1 ,x 2 )=
∑
w 1 ≤x 1
{
∑
x 2 <∞
pX 1 ,X 2 (w 1 ,x 2 )
}
.
By the uniqueness of cdfs, the quantity in braces must be the pmf ofX 1 evaluated
atw 1 ;thatis,
pX 1 (x 1 )=
∑
x 2 <∞
pX 1 ,X 2 (x 1 ,x 2 ), (2.1.8)
for allx 1 ∈DX 1. Hence, to find the probability thatX 1 isx 1 , keepx 1 fixed and
sumpX 1 ,X 2 over all ofx 2. In terms of a tabled joint pmf with rows comprised of
X 1 support values and columns comprised ofX 2 support values, this says that the
distribution ofX 1 can be obtained by the marginal sums of the rows. Likewise, the
pmf ofX 2 can be obtained by marginal sums of the columns.
Consider the joint discrete distribution of the random vector (X 1 ,X 2 )aspre-
sented in Example 2.1.1. In Table 2.1.1, we have added these marginal sums. The
final row of this table is the pmf ofX 2 , while the final column is the pmf ofX 1.
In general, because these distributions are recorded in the margins of the table, we
often refer to them asmarginalpmfs.
Example 2.1.4.Consider a random experiment that consists of drawing at random
one chip from a bowl containing 10 chips of the same shape and size. Each chip has
an ordered pair of numbers on it: one with (1,1), one with (2,1), two with (3,1),
one with (1,2), two with (2,2), and three with (3,2). Let the random variables
X 1 andX 2 be defined as the respective first and second values of the ordered pair.
Thus the joint pmfp(x 1 ,x 2 )ofX 1 andX 2 can be given by the following table, with
p(x 1 ,x 2 ) equal to zero elsewhere.