Page 157 - Probability and Statistical Inference

134 3. Multivariate Random Variables

Again note that with , the expression bh(x ) obtained from
2
(3.6.5) happens to be the pdf of a normal variable with mean µ and variance
2
at the point x ∈ ℜ. Hence, we must have
2

so that (3.6.8) can be rewritten as

by the definition of c from (3.6.2). Thus, we have directly verified that the
function f(x , x ) given by (3.6.1) is indeed a genuine pdf of a two-dimen-
1
2
sional random variable with its support ℜ .
2
Theorem 3.6.1 Suppose that (X , X ) has the N (µ , µ , , ρ)
2
1
distribution with its pdf f(x , x ) given by (3.6.1). Then, 2 1 2
1 2
(i) the marginal distribution of X is given by N(µ , ), i = 1, 2;
i i
(ii) the conditional distribution of X | X = x is normal with mean
1 2 2
µ + (x – µ ) and variance ), for all
1 2 2
fixed x ∈ℜ;
2
(iii) the conditional distribution of X | X = x is normal with
1
1
2
mean µ + (x – µ ) and variance , for all fixed
1
1
2
x ∈ℜ.
1
Proof (i) We simply show the derivation of the marginal pdf of the ran-
dom variable X . Using (3.3.2) one gets for any fixed x ∈ ℜ,
2 2

which can be expressed as

This shows that X is distributed as N(µ , ). The marginal pdf of X can be
2
2
1
found easily by appropriately modifying (3.6.5) first. We leave this as the
Exercise 3.6.1.