Page 161 - Probability and Statistical Inference

138 3. Multivariate Random Variables

that

whatever be fixed x ∈ ℜ, since α = .5. Suppose that it is possible for the pair
2
(X , X ) to be distributed as the bivariate normal variable, N (0,0,1,1, ρ*) with
1 2 2
some ρ* ∈ (–1, 1). But, then E[X | X = x ] must be ρ*x which has to match
1 2 2 2
with the answer zero obtained in (3.6.19), for all x ∈ ℜ. In other words, ρ*
2
2
must be zero. Hence, for all (x , x ) ∈ ℜ we should be able to write
1 2

Now, g(0,0; ρ = .5) = , but h(0,0) = 1/2π. It is obvious that g(0,0; π =.5)
≠ h(0,0). Hence, it is impossible for the random vector (X , X ) having the pdf
1 2
g(x , x ; ρ = .5) to be matched with any bivariate normal random vector. !
1 2
The Exercise 3.6.8 gives another pair of random variables X and
1
X such that marginally each is normally distributed whereas
2
jointly (X , X ) is not distributed as N .
1 2 2
For the sake of completeness, we now formally define what is known as
the regression function in statistics.
Definition 3.6.1 Suppose that (X , X ) has the N (µ , µ , , ρ)
1 2 2 1 2
distribution with its pdf f(x , x ) given by (3.6.1). The conditional mean of X
1 2 1
| X = x , that is, µ + (x –µ ) is known as the regression function of X on
2 2 1 2 2 1
X . Analogously, the conditional mean of X | X = x , that is, µ + (x – µ )
2 2 1 1 2 1 1
is known as the regression function of X on X .
2 1
Even though linear regression analysis is out of scope for this textbook,
we simply mention that it plays a very important role in statistics. The readers
have already noted that the regression functions in the case of a bivariate
normal distribution turn out to be straight lines.

We also mention that Tong (1990) had written a whole book devoted en-
tirely to the multivariate normal distribution. It is a very valuable resource,
particularly because it includes the associated tables for the percentage points
of the distribution.