Page 136 - Applied statistics and probability for engineers

114 Chapter 4/Continuous Random Variables and Probability Distributions

⎧0 x < −2 (d) Determine the cumulative distribution function and use the
⎪ cumulative distribution function to determine the probabil-
F x ( ) = ⎨ 0 .25 x + . 0 5 − ≤ x < 2
2
⎪ ≤ ity that you arrive between 8:15 a.m. and 8:30 a.m.
⎩ 1 2 x 4-27. The gap width is an important property of a magnetic
Determine the following: recording head. In coded units, if the width is a continuous ran-
(
(a) P X <1 8) (b) P X > − . ) 5 dom variable over the range from 0 < x < 2 with f x ( ) = 0 5. x,
(
.
1
(
P −
<
(c) P X < − ) 2 (d) ( 1< X ) 1 determine the cumulative distribution function of the gap width.
Determine the probability density function for each of the
4-19. Determine the cumulative distribution function for
following cumulative distribution functions.
the distribution in Exercise 4-1.
4-20. Determine the cumulative distribution function for F x ( ) = − − 2 x x >
4-28. 1 e 0
the distribution in Exercise 4-2.
4-29.
4-21. Determine the cumulative distribution function for the x <
⎧0 0
distribution in Exercise 4-3. ⎪ ≤ x <
⎪
4-22. Determine the cumulative distribution function for the F x ( ) = ⎨ . 0 2 x 0 ≤ x < 4
distribution in Exercise 4-4. ⎪ . 0 04 x + .0 64 4 9
⎪ 1 9 ≤ x
⎩
4-23. Determine the cumulative distribution function for the
4-30.
distribution in Exercise 4-5.
⎧0 x < −2
4-24. Determine the cumulative distribution function for the ⎪ − ≤ <
⎪
distribution in Exercise 4-8. Use the cumulative distribution F x ( ) = ⎨ . 0 25 x + . 0 5 2 x 1
<
0
function to determine the probability that a component lasts ⎪ . 0 5 x + .25 1 ≤ x 1 .5
⎩
5
more than 3000 hours before failure. ⎪ 1 1 . ≤ x
4-25. Determine the cumulative distribution function for the 4-31. Determine the cumulative distribution function for the
distribution in Exercise 4-11. Use the cumulative distribution random variable in Exercise 4-13.
function to determine the probability that a length exceeds 4-32. Determine the cumulative distribution function for the
2.7 meters. random variable in Exercise 4-14. Use the cumulative distri-
bution function to determine the probability that the random
4-26. The probability density function of the time you arrive
at a terminal (in minutes after 8:00 a.m.) is f x ( ) = 0 1. exp( 0.1− x) variable is less than 55.
4-33. Determine the cumulative distribution function for the
for 0 < x. Determine the probability that
random variable in Exercise 4-15. Use the cumulative distribu-
(a) You arrive by 9:00 a.m. tion function to determine the probability that 40 < X ≤ 60.
(b) You arrive between 8:15 a.m. and 8:30 a.m. 4-34. Determine the cumulative distribution function for the
(c) You arrive before 8:40 a.m. on two or more days of ive random variable in Exercise 4-16. Use the cumulative distribu-
days. Assume that your arrival times on different days are tion function to determine the probability that the waiting time
independent. is less than one hour.
4-4 Mean and Variance of a Continuous
Random Variable
The mean and variance can also be deined for a continuous random variable. Integration
replaces summation in the discrete deinitions. If a probability density function is viewed as a
loading on a beam as in Fig. 4-1, the mean is the balance point.
Mean and Variance
Suppose that X is a continuous random variable with probability density function
f x ( ). The mean or expected value of X, denoted as μ or E X ( ), is
∞
X
E
μ = ( ) = ∫ xf ( ) x dx (4-4)
−∞
2
The variance of X, denoted as V X ( ) or σ , is
∞ 2 ∞
μ
2
2
2
V X
f
s = ( ) = ( ∫ x − ) ( ) x dx = ∫ x f ( ) x dx − μ
−∞ −∞
2
The standard deviation of X is σ = σ .