Page 139 - Probability and Statistical Inference

116 3. Multivariate Random Variables

multidimensional random vector X by extending the essential ideas from (3.3.7)-
(3.3.8).
If one has a k-dimensional random variable X = (X , ..., X ), the joint pdf
k
1
would then be written as f(x) or f(x , ..., x ). The joint pdf of any subset of
k
1
random variables, for example, X and X , would then be found by integrating
1
3
f(x , ..., x ) with respect to the remaining variables x , x , ..., x . One can also
k
2
4
1
k
write down the expressions of the associated conditional pdf’s of any subset
of random variables from X given the values of any other subset of random
variables from X.
Theorem 3.3.2 Let X = (X , ..., X ) be any k-dimensional discrete or
1
k
continuous random variable. Suppose that we also have real valued functions
h (x) and constants a , i = 0, 1, ..., p. Then, we have
i i
as long as all the expectations involved are finite. That is, the expectation is
a linear operation.
Proof Let us write and hence we have

Now, the proof is complete. ¢
Next, we provide two specific examples.
Example 3.3.10 Let us denote χ = χ = (0, 1), χ = (0, 2) and define
1 3 2

Note that these are non-negative functions and ∫ a (x )dx = 1 for all
χi i i i