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5-6 BIVARIATE NORMAL DISTRIBUTION 177

Thus,

E1XY2 E1X2E1Y2 32 9 14 3218 32 0

It can be shown that these two random variables are independent. You can check that
f XY (x, y) f X (x) f Y (y) for all x and y.

However, if the correlation between two random variables is zero, we cannot immediately
conclude that the random variables are independent. Figure 5-13(d) provides an example.

EXERCISES FOR SECTION 5-5

5-67. Determine the covariance and correlation for the 5-73. Determine the value for c and the covariance and cor-
following joint probability distribution: relation for the joint probability density function f XY (x, y) c
over the range 0 x 5, 0 y, and x 1 y x 1.
x 1 1 2 4 5-74. Determine the covariance and correlation for the joint
y 3 4 5 6 probability density function f XY (x, y) 6 10 e
6 0.001x 0.002y
f XY (x, y) 1 8 1 4 1 2 1 8
over the range 0 x and x y from Example 5-15.
5-68. Determine the covariance and correlation for the 5-75. Determine the covariance and correlation for the joint
following joint probability distribution: probability density function f XY 1x, y2 e x y over the range
0 x and 0 y.
x 1 0.5 0.5 1 5-76. Suppose that the correlation between X and Y is . For
y 2 1 1 2
constants a, b, c, and d, what is the correlation between the
f XY (x, y) 1 8 1 4 1 2 1 8 random variables U aX b and V cY d?
5-69. Determine the value for c and the covariance and 5-77. The joint probability distribution is
correlation for the joint probability mass function f XY (x, y)
x 1 0 0 1
c(x y) for x 1, 2, 3 and y 1, 2, 3.
y 0 1 1 0
5-70. Determine the covariance and correlation for the joint
f XY (x, y) 1 4 1 4 1 4 1 4
probability distribution shown in Fig. 5-4(a) and described in
Example 5-8.
Show that the correlation between X and Y is zero, but X and Y
5-71. Determine the covariance and correlation for X 1 and are not independent.
X 2 in the joint distribution of the multinomial random vari- 5-78. Suppose X and Y are independent continuous random
1
ables X 1 , X 2 and X 3 in with p 1 p 2 p 3 3 and n 3. What
variables. Show that XY 0.
can you conclude about the sign of the correlation between
two random variables in a multinomial distribution?
5-72. Determine the value for c and the covariance and cor-
relation for the joint probability density function f XY (x, y)
cxy over the range 0 x 3 and 0 y x.

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