Applied Statistics and Probability for Engineers

178 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

chapter, if the specifications for Xand Yare 2.95 to 3.05 and 7.60 to 7.80 millimeters, respec-

tively, we might be interested in the probability that a part satisfies both specifications; that is,

P(2.95X3.05, 7.60Y7.80).

The probability density function of a bivariate normal distribution is

(5-32)

for and , with parameters X 0, Y 0,

  X 

,



  Y 

, and 1 1.



x

 

y



2  1 xX 21 yY 2

X Y 

1 yY 22

^2 Y

df

fXY 1 x, y; X, Y, X, Y,  2 

1

2 X Y 21 ^2

exp e

 1

211 ^22

c

1 xX 22

^2 X

Definition

Figure 5-17 Examples of bivariate normal distributions.

The result that fXY(x, y; X, Y, X, Y, ) integrates to 1 is left as an exercise. Also, the bivari-

ate normal probability density function is positive over the entire plane of real numbers.

Two examples of bivariate normal distributions are illustrated in Fig. 5-17 along with

corresponding contour plots. Each curve on the contour plots is a set of points for which the

probability density function is constant. As seen in the contour plots, the bivariate normal

probability density function is constant on ellipses in the (x,y) plane. (We can consider a circle

to be a special case of an ellipse.) The center of each ellipse is at the point (X, Y). If 

 0

(0), the major axis of each ellipse has positive (negative) slope, respectively. If 0, the

major axis of the ellipse is aligned with either the xor ycoordinate axis.

EXAMPLE 5-33 The joint probability density function is a special case of a bivariate

normal distribution with X1, Y1, X0, Y0, and 0. This probability density

function is illustrated in Fig. 5-18. Notice that the contour plot consists of concentric circles about

the origin.

By completing the square in the exponent, the following results can be shown. The details

are left as an exercise.

fXY 1 x, y 2 

1

12 

e0.5^1 x

(^2) y (^22)
x
y
fXY(x, y)
fXY(x, y)
y
x
0 x
y
y
x

Y
X
X
X
X

Y
Y

Y
fXY(x, y)
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