Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

86 Multivariate Distributions

with random variables in Section 1.5 we can uniquely definePX 1 ,X 2 in terms of the
cumulative distribution function(cdf), which is given by

FX 1 ,X 2 (x 1 ,x 2 )=P[{X 1 ≤x 1 }∩{X 2 ≤x 2 }], (2.1.1)

for all (x 1 ,x 2 )∈R^2. BecauseX 1 andX 2 are random variables, each of the events
in the above intersection and the intersection of the events are events in the original
sample spaceC. Thus the expression is well defined. As with random variables, we
writeP[{X 1 ≤x 1 }∩{X 2 ≤x 2 }]asP[X 1 ≤x 1 ,X 2 ≤x 2 ]. As Exercise 2.1.3 shows,

P[a 1 <X 1 ≤b 1 ,a 2 <X 2 ≤b 2 ]=FX 1 ,X 2 (b 1 ,b 2 )−FX 1 ,X 2 (a 1 ,b 2 )

−FX 1 ,X 2 (b 1 ,a 2 )+FX 1 ,X 2 (a 1 ,a 2 ).(2.1.2)

Hence, all induced probabilities of sets of the form (a 1 ,b 1 ]×(a 2 ,b 2 ] can be formulated
in terms of the cdf. We often call this cdf thejoint cumulative distribution
function of (X 1 ,X 2 ).
As with random variables, we are mainly concerned with two types of random
vectors, namely discrete and continuous. We first discuss the discrete type.
A random vector (X 1 ,X 2 )isadiscrete random vectorif its spaceDis finite
or countable. Hence,X 1 andX 2 are both discrete also. Thejoint probability
mass function(pmf) of (X 1 ,X 2 ) is defined by

pX 1 ,X 2 (x 1 ,x 2 )=P[X 1 =x 1 ,X 2 =x 2 ], (2.1.3)

for all (x 1 ,x 2 )∈D. As with random variables, the pmf uniquely defines the cdf. It
also is characterized by the two properties

(i) 0≤pX 1 ,X 2 (x 1 ,x 2 )≤1 and (ii)

∑∑

D

pX 1 ,X 2 (x 1 ,x 2 )=1. (2.1.4)

For an eventB∈D,wehave

P[(X 1 ,X 2 )∈B]=

∑∑

B

pX 1 ,X 2 (x 1 ,x 2 ).

Example 2.1.1.Consider the example at the beginning of this section where a fair
coin is flipped three times andX 1 andX 2 are the number of heads on the first two
flips and all 3 flips, respectively. We can conveniently table the pmf of (X 1 ,X 2 )as

Support ofX 2

0123

0 18 18 00

Support ofX 1 1 0 28 28 0

(^2001818)
For instance,P(X 1 ≥ 2 ,X 2 ≥2) =p(2,2) +p(2,3) = 2/8.