Page 93 - Applied Statistics And Probability For Engineers

c03.qxd 8/6/02 2:41 PM Page 71

3-5 DISCRETE UNIFORM DISTRIBUTION 71

EXAMPLE 3-14 As in Example 3-1, let the random variable X denote the number of the 48 voice lines that are
in use at a particular time. Assume that X is a discrete uniform random variable with a range
of 0 to 48. Then,

E1X2 148 02 2 24

and
2
53148 0 12 14 126 1 2 14.14
Equation 3-6 is more useful than it might first appear. If all the values in the range of a
random variable X are multiplied by a constant (without changing any probabilities), the mean
and standard deviation of X are multiplied by the constant. You are asked to verify this result
in an exercise. Because the variance of a random variable is the square of the standard devia-
tion, the variance of X is multiplied by the constant squared. More general results of this type
are discussed in Chapter 5.

EXAMPLE 3-15 Let the random variable Y denote the proportion of the 48 voice lines that are in use at a par-
ticular time, and X denotes the number of lines that are in use at a particular time. Then,
Y X 48 . Therefore,

E1Y2 E1X2 48 0.5

and
2
V1Y2 V1X2 48 0.087

EXERCISES FOR SECTION 3-5

3-46. Let the random variable X have a discrete uniform 590.0 and continuing through 590.9. Determine the mean and
distribution on the integers 0 x 100 . Determine the mean variance of lengths.
and variance of X. 3-52. Suppose that X has a discrete uniform distribution on
3-47. Let the random variable X have a discrete uniform the integers 0 through 9. Determine the mean, variance, and
distribution on the integers 1 x 3 . Determine the mean standard deviation of the random variable Y 5X and com-
and variance of X. pare to the corresponding results for X.
3-48. Let the random variable X be equally likely to assume 3-53. Show that for a discrete uniform random variable X,
any of the values 1 8 , 1 4 , or 3 8 . Determine the mean and if each of the values in the range of X is multiplied by the
variance of X. constant c, the effect is to multiply the mean of X by c and
3-49. Thickness measurements of a coating process are the variance of X by c 2 . That is, show that E1cX 2 cE1X 2
2
made to the nearest hundredth of a millimeter. The thickness and V1cX 2 c V1X 2 .
measurements are uniformly distributed with values 0.15, 3-54. The probability of an operator entering alphanu-
0.16, 0.17, 0.18, and 0.19. Determine the mean and variance meric data incorrectly into a field in a database is equally
of the coating thickness for this process. likely. The random variable X is the number of fields on a
3-50. Product codes of 2, 3, or 4 letters are equally likely. data entry form with an error. The data entry form has
What is the mean and standard deviation of the number of 28 fields. Is X a discrete uniform random variable? Why or
letters in 100 codes? why not.
3-51. The lengths of plate glass parts are measured to the
nearest tenth of a millimeter. The lengths are uniformly dis-
tributed, with values at every tenth of a millimeter starting at