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5-6 BIVARIATE NORMAL DISTRIBUTION 179

f XY (x, y)
y
z
y
y
0
0

0 0 x
x x
Figure 5-18 Bivariate normal probability density Figure 5-19 Marginal probability
function with X 1, Y 1, 0, X 0, and density functions of a bivariate
Y 0. normal distribution.

Marginal
Distributions of If X and Y have a bivariate normal distribution with joint probability density f (x, y;
XY
Bivariate Normal , , , , ), the marginal probability distributions of X and Y are normal
Random Variables X Y X Y
with means and and standard deviations and , respectively. (5-33)
Y
X
Y
X

Figure 5-19 illustrates that the marginal probability distributions of X and Y are normal.
Furthermore, as the notation suggests, represents the correlation between X and Y. The
following result is left as an exercise.

If X and Y have a bivariate normal distribution with joint probability density function
f (x, y; , , , , ), the correlation between X and Y is . (5-34)
X
Y
Y
X
XY
The contour plots in Fig. 5-17 illustrate that as moves from zero (left graph) to 0.9 (right
graph), the ellipses narrow around the major axis. The probability is more concentrated about
a line in the (x, y) plane and graphically displays greater correlation between the variables. If
1 or 1, all the probability is concentrated on a line in the (x, y) plane. That is, the
probability that X and Y assume a value that is not on the line is zero. In this case, the bivari-
ate normal probability density is not defined.
In general, zero correlation does not imply independence. But in the special case that X
and Y have a bivariate normal distribution, if 0, X and Y are independent. The details are
left as an exercise.

If X and Y have a bivariate normal distribution with 0, X and Y are independent.
(5-35)