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3. Multivariate Random Variables 115

to the marginal pdf f (x ) of X from (3.3.10), we get
1 1 1

which matches with the answer found earlier in (3.3.12). Next, combining
(3.3.15) and (3.3.17) with the Theorem 3.3.1 (ii), we should then have

Now, we combine (3.3.39) and (3.3.40). We also use the marginal pdf f (x )
1
1
of X from (3.3.10) and direct integration to write
1

2
since E[(1 – X )] = 3/4 and E[(1 – X ) ] = 3/5. The answer given earlier in
1 1
(3.3.13) and the one found in (3.3.41) match exactly. !
3.3.2 Three and Higher Dimensions

The ideas expressed in (3.3.2) and (3.3.5)-(3.3.6) extend easily in a multi-
variate situation too. For example, let us suppose that a random vector X =
(X , X , X , X ) has the joint pdf f(x , x , x , x ) where x ∈ χ , the support of
1
3
2
i
4
1
4
i
2
3
X , i = 1, 2, 3, 4. The marginal pdf of (X , X ), that is the joint pdf of X and
3
1
i
1
X , for example, can be found as follows:
3
The conditional pdf of (X , X ) given that X = x , X = x will be of the form
2 4 1 1 3 3
and the x’s belong to the χ’s. The notions of expectations and conditional ex-
pectations can also be generalized in a natural fashion in the case of a