Page 101 - Applied statistics and probability for engineers

Section 3-5/Discrete Uniform Distribution 79

0
1
σ = (48 − + ) 2 −1 = 14 .14
12
Practical Interpretation: The average number of lines in use is 24, but the dispersion (as measured by σ) is large.
Therefore, at many times far more or fewer than 24 lines are used.

Equation 3-6 is more useful than it might irst appear. For example, suppose that the discrete uni-
form random variable Y has range 5 10, ,… , 30. Then Y = 5 X where X has range 1 2,… 6 , . The mean
,
and variance of Y are obtained from the formulas for a linear function of X in Section 3-4 to be
⎛1 + ⎞ 6
E Y) = 5 E X) = 5 ⎜ ⎟ = 17 .5
(
(
⎝ 2 ⎠
2
⎡ 6 1 ) − ⎤ 1
( − +1
2
(
.
V Y) = 5 V X) = 25 ⎢ ⎥ ⎥ = 72 92
(
⎣ 12 ⎦
Example 3-15 Proportion of Voice Lines Let the random variable Y denote the proportion of the 48 voice lines used at
a particular time, and X denote the number of lines used at a particular time. Then Y = X / 48. Therefore,
E Y ( ) = ( = .5
E X) / 48
0
and
V Y ( ) = ( 2 = .087
V X) / 48
0
Exercises FOR SECTION 3-5
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
3-76. Let the random variable X have a discrete uniform dis- and variance of the wavelength distribution compare to the
tribution on the integers 0 ≤ ≤x 99. Determine the mean and previous part? Explain.
variance of X. 3-82. The probability of an operator entering alphanu-

3-77. Let the random variable X have a discrete uniform meric data incorrectly into a ield in a database is equally likely.
distribution on the integers 1≤ ≤x 3. Determine the mean and The random variable X is the number of i elds on a data entry
variance of X. form with 28 i elds that have an error. Is X a discrete uniform
3-78. Thickness measurements of a coating process are random variable? Why or why not?
made to the nearest hundredth of a millimeter. The thickness 3-83. Suppose that X has a discrete uniform distribution
measurements are uniformly distributed with values 0.15, 0.16, on the integers 0 through 9. Determine the mean, variance,
0.17, 0.18, and 0.19. Determine the mean and variance of the and standard deviation of the random variable Y = 5 X and
coating thickness for this process. compare to the corresponding results for X.
3-79. Product codes of two, three, four, or i ve letters are 3-84. Show that for a discrete uniform random variable X, if each
equally likely. What are the mean and standard deviation of the of the values in the range of X is multiplied by the constant c, the
2
number of letters in the codes? effect is to multiply the mean of X by c and the variance of X by c .
2
cE X
3-80. The lengths of plate glass parts are measured to the That is, show that E cX ( ) = ( ) and V cX ( ) = c V X ( ).
nearest tenth of a millimeter. The lengths are uniformly dis- 3-85. The number of pages in a PDF document you create has
tributed with values at every tenth of a millimeter starting at a discrete uniform distribution from i ve to nine pages (includ-
590.0 and continuing through 590.9. Determine the mean and ing the end points). What are the mean and standard deviation
variance of the lengths. of the number of pages in the document?
3-81. Assume that the wavelengths of photosynthetically 3-86. Suppose that nine-digit Social Security numbers are
active radiations (PAR) are uniformly distributed at integer assigned at random. If you randomly select a number, what is
nanometers in the red spectrum from 675 to 700 nm. the probability that it belongs to one of the 300 million people
(a) What are the mean and variance of the wavelength distribu- in the United States?
tion for this radiation? 3-87. Suppose that 1000 seven-digit telephone numbers
(b) If the wavelengths are uniformly distributed at integer within your area code are dialed randomly. What is the prob-
nanometers from 75 to 100 nanometers, how do the mean ability that your number is called?