Page 141 - Probability and Statistical Inference

118 3. Multivariate Random Variables

Utilizing (3.3.49), it also follows that for any 0 < x < 1, the marginal pdf of X 3
3
is given by

Utilizing the expressions of g(x , x ), g (x ) from (3.3.49) and (3.3.51), for
3
3
3
1
any 0 < x < 1, we can write down the conditional pdf of X given that X = x 3
1
1
3
as follows:
After obtaining the expression of g(x , x ) from (3.3.49), one can easily
1 3
evaluate, for example E(X X ) or E(X ) respectively as the double integrals
1 3 1
. Look
at the Exercises 3.3.6-3.3.7. !
In the two Examples 3.3.10-3.3.11, make a special note of how
the joint densities f(x) of more than two continuous random
variables have been constructed. The readers should create a
few more of their own. Is there any special role of the specific
forms of the functions a (x )’s? Look at the Exercise 3.3.8.
i i
Example 3.3.11 Let us denote χ = χ = χ = χ = (0, 1), χ = (0, 2) and
1
2
3
4
5
recall the functions a (x ), a (x ) and a (x ) from (3.3.45)-(3.3.46). Addition-
3
1
1
2
3
2
ally, let us denote
Note that these are non-negative functions and also one has ∫ a (x )dx = 1 for
χi i i i
all i = 1, ..., 5. With x = (x , x , x , x , x ) and , let us denote
1 2 3 4 5

It is a fairly simple matter to check that ,