Applied Statistics and Probability for Engineers

5-6 BIVARIATE NORMAL DISTRIBUTION 179

Figure 5-18 Bivariate normal probability density

function withX 1 , Y 1 , 0,X0, and

Y0.

0

0

x

y

fXY(x, y)

(^0) x
y
0
If Xand Yhave a bivariate normal distribution with joint probability density fXY(x,y;
X, Y, X, Y, ), the marginal probability distributionsof Xand Yare normal
with meansXand Yand standard deviations Xand Y, respectively. (5-33)
Marginal
Distributions of
Bivariate Normal
Random Variables
If Xand Yhave a bivariate normal distribution with joint probability density function
fXY(x, y; X, Y, X, Y, ), the correlation between Xand Yis . (5-34)
x
z
y
Figure 5-19 Marginal probability
density functions of a bivariate
normal distribution.
Figure 5-19 illustrates that the marginal probability distributions of Xand Yare normal.
Furthermore, as the notation suggests, represents the correlation between Xand Y. The
following result is left as an exercise.
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 or1, all the probability is concentrated on a line in the (x,y) plane. That is, the
probability that Xand Yassume 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 Yhave a bivariate normal distribution, if 0, Xand Yare independent. The details are
left as an exercise.
If Xand Yhave a bivariate normal distribution with 0, Xand Yare independent.
(5-35)
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