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Section 5-3/Common Joint Distributions 179

denote the number of patients who improve or degrade. Are 5-42. Determine the covariance and correlation for the joint
X and Y independent? Calculate the covariance and correla- probability density function f XY ( x, y) = 6 10× −6 e −0.001 x − 0.002 y
tion between X and Y . over the range 0 < x and x < y from Example 5-2.
5-38. For the Transaction Processing Performance Council’s 5-43. The joint probability distribution is
,
Z
benchmark in Exercise 5-10, let X, Y and denote the average x –1 0 0 1
number of selects, updates, and inserts operations required for y 0 –1 1 0
each type of transaction, respectively. Calculate the following: f XY ( x, y) 1 4 1 4 1 4 1 4
/
/
/
/
(a) Covariance between X and Y
(b) Correlation between X and Y Show that the correlation between X and Y is zero but X and Y
(c) Covariance between X and Z are not independent.
(d) Correlation between X and Z 5-44. Determine the covariance and correlation for the
5-39. Determine the value for c and the covariance CD4 counts in a month and the following month in Exer-
and correlation for the joint probability density function cise 5-30.
f XY ( ) = 3 and 0 < < x. 5-45. Determine the covariance and correlation for the lengths
y
x, y cxy over the range 0 < <x
5-40. Determine the value for c and the covariance and cor- of the minor and major axes in Exercise 5-29.
relation for the joint probability density function f XY ( x, y) = c 5-46. Suppose that X and Y are independent continuous random
over the range 0 < <x 5, 0 < y, and x −1< < x + 1. variables. Show that σ XY = 0.
y
5-41. Determine the covariance and correlation for the 5-47. Suppose that the correlation between X and Y is ρ .
−
x, y
joint probability density function f XY ( ) = e − x y over the For constants a, b, c, and d, what is the correlation between
range 0 < x and 0 < y. the random variables U = aX + b and V = cY + d?
5-3 Common Joint Distributions
5-3.1 MULTINOMIAL PROBABILITY DISTRIBUTION
The binomial distribution can be generalized to generate a useful joint probability distribution
for multiple discrete random variables. The random experiment consists of a series of inde-
pendent trials. However, the outcome from each trial is categorized into one of k classes. The
random variables of interest count the number of outcomes in each class.

Example 5-21 Digital Channel We might be interested in a probability such as the following. Of the 20 bits
received, what is the probability that 14 are excellent, 3 are good, 2 are fair, and 1 is poor? Assume that
the classiications of individual bits are independent events and that the probabilities of E, G, F, and P are 0.6, 0.3, 0.08, and

0.02, respectively. One sequence of 20 bits that produces the speciied numbers of bits in each class can be represented as
EEEEEEEEEEEEEEGGGFFP

Using independence, we ind that the probability of this sequence is
( 14 3 2 1
P EEEEEEEEEEEEEEGGGFFP) = .6 0 .3 0 .08 0 .02
0
×
= .708 10 −9
2
Clearly, all sequences that consist of the same numbers of E’s, G’s, F’s, and P’s have the same probability. Conse-
−
9
×
quently, the requested probability can be found by multiplying 2.708 10 by the number of sequences with 14 E’s,
3 G’s, 2 F’s, and 1 P. The number of sequences is found from Chapter 2 to be
20!
!
!
14 3 2 1! = 2,325,600
!
Therefore, the requested probability is
. (
( P) = 2325600 2 708 10 ) = .
−
×
9
s
P 14 E , G , F ,and1 0 0063
’
2
’
s
3
’
s
Example 5-24 leads to the following generalization of a binomial experiment and a
binomial distribution.