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110 3. Multivariate Random Variables
for 0 < x < 1. Next, we can simply use the marginal pdf f (x ) of X to
2
2
2
2
obtain
so that
The reader may similarly evaluate E(X ) and V(X ). !
1 1
Example 3.3.2 (Example 3.3.1 Continued) One should easily verify that
are respectively the two conditional pdfs. For fixed 0 < x < 1, the conditional
1
expectation of X given that X = x is obtained as follows:
2 1 1
One can evaluate the expression of as follows:
so that using (3.3.14) and after considerable simplifications, we find:
The details are left out as Exercise 3.3.1.!
Example 3.3.3 Consider two random variables X and X whose joint con-
tinuous distribution is given by the following pdf: 1 2
Obviously, one has χ = χ = (0, 1). One may check the following expressions
1
2
for the marginal pdfs easily:
