Page 138 - Applied statistics and probability for engineers

116 Chapter 4/Continuous Random Variables and Probability Distributions

Exercises FOR SECTION 4-4

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-35. Suppose that f x ( ) = .25 for 0 < < 4. Determine (a) Determine the mean and standard deviation of the cable
x
0
the mean and variance of X. length.
4-36. Suppose that f x ( ) = .125 x for 0 < < 4. Deter- (b) If the length speciications are 1195 < <x 1205 millimeters,
0
x
mine the mean and variance of X. what proportion of cables is within speciications?
2
1
. 1
x
x
4-37. Suppose that f x ( ) = .5 for −1< < Determine 4-47. The thickness of a conductive coating in microm-
−
μ
2
x
the mean and variance of X. eters has a density function of 600x for 100 mμ < < 120 m.
x
x
4-38. Suppose that f x ( ) = /8 for 3 < < 5. Determine (a) Determine the mean and variance of the coating thickness.
the mean and variance of x. (b) If the coating costs $0.50 per micrometer of thickness on
4-39. Determine the mean and variance of the random variable each part, what is the average cost of the coating per part?
in Exercise 4-1. 4-48. The probability density function of the weight of
x )
( /
4-40. Determine the mean and variance of the random variable packages delivered by a post ofice is f x ( ) = 70 69 2 for
x
in Exercise 4-2. 1< < 70 pounds.
4.41 Determine the mean and variance of the random variable (a) Determine the mean and variance of weight.
in Exercise 4-13. (b) If the shipping cost is $2.50 per pound, what is the average
4.42 Determine the mean and variance of the random variable shipping cost of a package?
in Exercise 4-14. (c) Determine the probability that the weight of a package
4.43 Determine the mean and variance of the random variable exceeds 50 pounds.
in Exercise 4-15. 4-49. Integration by parts is required. The probability
4.44 Determine the mean and variance of the random variable density function for the diameter of a drilled hole in millim-
5)
− (
10 x −
in Exercise 4-16. eters is 10e for x > 5 mm. Although the target diameter
4-45. Suppose that contamination particle size (in microm- is 5 millimeters, vibrations, tool wear, and other nuisances pro-
eters) can be modeled as f x ( ) = 2 x −3 for 1< x. Determine the duce diameters greater than 5 millimeters.
mean of X. What can you conclude about the variance of X? (a) Determine the mean and variance of the diameter of the holes.
4-46. Suppose that the probability density function of (b) Determine the probability that a diameter exceeds 5.1
the length of computer cables is f x ( ) = 0 1. from 1200 to 1210 millimeters.
millimeters.
4-5 Continuous Uniform Distribution
The simplest continuous distribution is analogous to its discrete counterpart.
Continuous Uniform
Distribution A continuous random variable X with probability density function
x
f x ( ) = ( 1 b − ) a ≤ ≤ b (4-6)
a ,
is a continuous uniform random variable.
The probability density function of a continuous uniform random variable is shown in Fig. 4-8.
The mean of the continuous uniform random variable X is
b x 2 b ( a + b)
E X ( ) = ∫ dx = 0. x 5 =
−
a b a b − a a 2
The variance of X is
b ⎛ + 2 ⎛ + 3 b
⌠ ⎜ x − ( a b ⎞ ) ⎟ ⎜ x − ( a b ⎞ ) ⎟
−
⎮ ⎝ 2 ⎠ ⎝ 2 ⎠ ( (b a ) 2
V X ( ) = ⎮ dx = =
−
⎮ b − a ( 3 b a) a 12
⎮
⌡
a