Page 132 - Probability and Statistical Inference

3. Multivariate Random Variables 109

Hence, the conditional pdf of X given that X = x is given by
1 2 2

Once these conditional pdf’s are obtained, one can legitimately start talk-
ing about the conditional moments of one of the random variables given the
other. For example, in order to find the conditional mean and the conditional
variance of X given that X = x , one should proceed as follows:
1 2 2

It should be clear that µ and respectively from (3.3.7) would corre-
1/2
spond to the conditional mean and the conditional variance of X given that
1
X = x . In general, we can write
2 2

Example 3.3.1 Consider two random variables X and X whose joint
1
2
continuous distribution is given by the following pdf:

Obviously, one has χ = χ = (0, 1). Next, we integrate this joint pdf with
2
1
respect to x to obtain the marginal pdf of X and write
2 1

for 0 < x < 1. In the same fashion, we obtain the marginal pdf of X as
2
1
follows: