Page 134 - Probability and Statistical Inference

3. Multivariate Random Variables 111

One should derive the conditional pdf’s and the associated conditional means
and variances. See the Exercise 3.3.2. !

In a continuous bivariate distribution, the joint, marginal, and
conditional pdf’s were defined in (3.3.1)-(3.3.2) and (3.3.5)-(3.3.6).

In the case of a two-dimensional random variable, anytime we wish to
2
calculate the probability of an event A(⊆ ℜ ), we may use an approach which
is a generalization of the equation (1.6.2) in the univariate case: One would
write

We emphasize that the convention is to integrate f(x , x ) only on that part of
the set A where f(x , x ) is positive. 1 2
1 2
If we wish to evaluate the conditional probability of an event B(⊆ ℜ)
given, say, X = x , then we should integrate the conditional pdf f (x ) of X 2
1
1
2/1
2
given X = x over that part of the set B where f (x ) is positive. That is, one
1
1
2/1
2
has
Example 3.3.4 (Example 3.3.2 Continued) In order to appreciate the
essence of what (3.3.20) says, let us go back for a moment to the Ex-
ample 3.3.2. Suppose that we wish to find the probability of the set or
the event A where A = {X = .2 ∩.3 < X = .8}. Then, in view of (3.3.20),
1 2
we obtain
dx = 1.2(.845 – .62) = .27. !
1
Example 3.3.5 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). One may easily check the following
1
2
expressions for the marginal pdf’s:

Now, suppose that we wish to compute the conditional probability that
X < .2 given that X = .5. We may proceed as follows. We have
2 1