Applied Statistics and Probability for Engineers

5-6 BIVARIATE NORMAL DISTRIBUTION 177

Thus,

It can be shown that these two random variables are independent. You can check that

fXY(x,y)fX(x)fY(y) for all xand y.

However, if the correlation between two random variables is zero, we cannotimmediately

conclude that the random variables are independent. Figure 5-13(d) provides an example.

EXERCISES FOR SECTION 5-5

E 1 XY 2 E 1 X 2 E 1 Y 2  32 9  14 3218 32  0

5-67. Determine the covariance and correlation for the

following joint probability distribution:

x 1124

y 3456

fXY(x, y)1 81  41  21  8

5-68. Determine the covariance and correlation for the

following joint probability distribution:

x  1 0.5 0.5 1

y  2  112

fXY(x,y)

5-69. Determine the value for cand the covariance and

correlation for the joint probability mass function fXY(x,y)

c(xy) for x1, 2, 3 and y1, 2, 3.

5-70. Determine the covariance and correlation for the joint

probability distribution shown in Fig. 5-4(a) and described in

Example 5-8.

5-71. Determine the covariance and correlation for X 1 and

X 2 in the joint distribution of the multinomial random vari-

ables X 1 , X 2 and X 3 in with p 1  p 2  p 3 ^1  3 and n 3. What

can you conclude about the sign of the correlation between

two random variables in a multinomial distribution?

5-72. Determine the value for cand the covariance and cor-

relation for the joint probability density function fXY(x,y)

cxyover the range 0x3 and 0yx.

1 8 1 4 1 2 1 8

5-6 BIVARIATE NORMAL DISTRIBUTION

An extension of a normal distribution to two random variables is an important bivariate prob-

ability distribution.

EXAMPLE 5-32 At the start of this chapter, the length of different dimensions of an injection-molded part was

presented as an example of two random variables. Each length might be modeled by a normal

distribution. However, because the measurements are from the same part, the random

variables are typically not independent. A probability distribution for two normal random vari-

ables that are not independent is important in many applications. As stated at the start of the

5-73. Determine the value for cand the covariance and cor-

relation for the joint probability density function fXY(x,y)c

over the range 0x5, 0y, and x 1 yx1.

5-74. Determine the covariance and correlation for the joint

probability density function fXY(x, y) 6 10 ^6 e0.001x0.002y

over the range 0xand xyfrom Example 5-15.

5-75. Determine the covariance and correlation for the joint

probability density function over the range

0 xand 0y.

5-76. Suppose that the correlation between Xand Yis . For

constants a, b, c, and d, what is the correlation between the

random variables U aXband VcYd?

5-77. The joint probability distribution is

x  101

y 11 0

fXY(x,y)

Show that the correlation between Xand Yis zero, but Xand Y

are not independent.

5-78. Suppose Xand Yare independent continuous random

variables. Show that XY0.

1 4 1 4 1 4 1 4

0

0

fXY 1 x, y 2 exy

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