Page 125 - Probability and Statistical Inference

102 3. Multivariate Random Variables

For i≠ j, the conditional pmf of X at the point x given that X = x , denoted
i i j j
by f (x ), is defined by
i|j i

Now, suppose that P(X = x ), which is equivalent to f (x ), is positive. Then,
j
j
j
j
using the notion of the conditional probability from (1.4.1) we can express
f (x ) as
i|j i
for x ∈ χ , i ≠ j. In the two-dimensional case, we have two conditional
i
i
pmf’s, namely f (x ) and f (x ) which respectively represent the conditional
2
1
2|1
1|2
pmf of X given X = x and that of X given X = x .
1 2 2 2 1 1
Once the conditional pmf f (x ) of X given X = x and the conditional
1
1|2
2
2
1
pmf f (x ) of X given X = x are found, the conditional mean of X given X j
i
2
1
2
2|1
1
= x , denoted by E[X | X = x ], is defined as follows:
j i j j
This is not really any different from how we interpreted the expected value of
a random variable in the Definition 2.2.1. To find the conditional expectation
of X given X = x , we simply multiply each value x by the conditional prob-
j
i
j
i
ability P{X = x | X = x } and then sum over all possible values x ∈ χ with any
i
i
i
j
j
i
fixed x ∈ χ , for i ≠ j = 1, 2.
j j
Example 3.2.3 (Example 3.2.2 Continued) One can verify that given X =
2
–1, the conditional pmf f (x ) corresponds to 0, 1 and 0 respectively when x 1
1|2
1
= 0, 1, 2. Other conditional pmf’s can be found analogously. !
Example 3.2.4 Consider two random variables X and X whose joint dis-
1
2
tribution is given as follows:
Table 3.2.2. Joint Distribution of X and X
1 2
X values Row Total
1
–1 2 5
1 .12 .18 .25 .55
X values
2
2 .20 .09 . 16 .45
Col. Total .32 .27 .41 1.00