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3

Multivariate Random Variables

3.1 Introduction
Suppose that we draw two random digits, each from the set {0, 1, ...,
9}, with equal probability. Let X , X be respectively the sum and the differ-
1
2
ence of the two selected random digits. Using the sample space technique,
one can easily verify that P(X = 0) = 1/100, P(X = 1) = 2/100, P(X = 2)
1
1
1
= 3/100 and eventually obtain the distribution, namely P(X = i) for i = 0, 1,
1
..., 18. Similarly, one can also evaluate P(X = j) for j = –9, –8, ..., 8, 9.
2
Now, suppose that one has observed X = 3. Then there is no chance for X
1 2
to take a value such as 2. In other words, the bivariate discrete random
variable (X , X ) varies together in a certain way. This sense of joint varia-
2
1
tion is the subject matter of the present chapter.
Generally speaking, we may have k(≥ 2) real valued random variables X ,
1
..., X which vary individually as well as jointly. For example, during the
k
“health awareness week” we may consider a population of college students
and record a randomly selected student’s height (X ), weight (X ), age (X )
1
3
2
and blood pressure (X ). Each individual random variable may be assumed
4
to follow some appropriate probability distribution in the population, but it
may also be quite reasonable to assume that these four variables together
follow a certain joint probability distribution. This example falls in the cat-
egory of multivariate continuous random variables.
The Section 3.2 introduces the ideas of joint, marginal and conditional
distributions in a discrete situation including the multinomial distribution.
The Section 3.3 introduces analogous topics in the continuous case. We
begin with the bivariate scenario for simplicity. The notion of a compound
distribution and the evaluation of the associated mean and variance are ex-
plored in the Examples 3.3.6-3.3.8. Multidimensional analysis is included in
the Section 3.3.2. The notions of the covariances and correlation coeffi-
cients between random variables are explored in the Section 3.4. We briefly
revisit the multinomial distribution in the Section 3.4.1. The Section 3.5
introduces the concept of the independence of random variables. The cel-
ebrated bivariate normal distribution is discussed in the Section 3.6 and the
associated marginal as well as the conditional pdf’s are found in this special
situation. The relationships between the zero correlation and possible
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