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3. Multivariate Random Variables 107

dimensional random variable (X , X , ) where ).By
1 5
the Theorem 3.2.2, the three dimensional variable of interest (X , X , )has
1 5
the Mult (n, 1/6, 1/6, 2/3) distribution where n = 20. !
3
3.3 Continuous Distributions
Generally speaking, we may be dealing with k(≥ 2) real valued random vari-
ables X , ..., X which vary individually as well as jointly. We may go back to
1
k
the example cited earlier in the introduction. Consider a population of college
students and record a randomly selected student’s height (X ), weight (X ),
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1
age (X ), and blood pressure (X ). It may be quite reasonable to assume that
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4
these four variables together has some multivariate continuous joint probabil-
ity distribution in the population.
The notions of the joint, marginal, and conditional distributions in the
case of multivariate continuous random variables are described in the Section
3.3.1. This section begins with the bivariate scenario for simplicity. The no-
tion of a compound distribution and the evaluation of the associated mean and
variance are explored in the Examples 3.3.6-3.3.8. Multidimensional analysis
is included in the Section 3.3.2.

3.3.1 The Joint, Marginal and Conditional Distributions
For the sake of simplicity, we begin with the bivariate continuous random
variables. Let X , X be continuous random variables taking values respec-
1
2
tively in the spaces χ , χ both being subintervals of the real line ℜ. Consider
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2
a function f(x , x ) defined for (x , x ) ∈ χ × χ . Throughout we will use the
2
1
2
1
2
1
following convention:
Let us presume that χ × χ is the support of the probability
1
2
distribution in the sense that f(x , x ) > 0 for all (x , x )
2
1
2
1
∈ χ × χ and f(x , x ) = for all (x , x ) ∈ℜ – (χ × χ ).
2
1 2 1 2 1 2 1 2
Such a function f(x , x ) would be called a joint pdf of the random vector
2
1
X = (X , X ) if the following condition is satisfied:
1 2
Comparing (3.3.1) with the requirement (ii) in (3.2.2) and (1.6.1), one
will notice obvious similarities. In the case of one-dimensional random vari-
ables, the integral in (1.6.1) was interpreted as the total area under the
density curve y = f(x). In the present situation involving two random