5-1
CHAPTER 5. PROBABILITY DISTRIBUTIONS FOR
CONTINUOUS RANDOM VARIABLES
CHAPTER 5. PROBABILITY DISTRIBUTIONS FOR CONTINUOUS RANDOM
VARIABLES …………………………………………………………………………………………………………………….. 1
5.1. Introduction ………………………………………………………………………………………………………………1
5.2. Uniform Distributions ………………………………………………………………………………………………..1
5.3. Normal Distributions ………………………………………………………………………………………………….2
5.4. Lognormal Distributions ……………………………………………………………………………………..……..4
The following table provides a summary of the problems with their appropriate sections:
Section Problems
5.2 1 to 3
5.3 4 to 8
5.1. Introduction
None
5.2. Uniform Distributions
Problem 5-1.
Using the uniform distribution with a = 0 and b = 0.35, the following probabilities can be
computed:
5-2
Problem 5-2.
Problem 5-3.
Range 50-59 60-69 70-74 75-79 80-84 85-89 90-94 95-100
Using the grade as a random variable X in the range 0 to 100, the uniform distribution is defined as
follow:
5.3. Normal Distributions
Problem 5-4.
Problem 5-5.
Problem 5-6.
5-3
(e) Find x such that P(X > x) = 0.05, or
Problem 5-7.
Problem 5-8.
1. A twoway trip X2
5-4
2. A complete two-way trip with unloading X2u
5.4. Lognormal Distributions
Problem 5-9.
Problem 5-10.
Lognormal distribution: P x = 1.45, Vx = 0.2
5-5
Problem 5-11.
Convert P x and Vx to parameters
Problem 5-12.
Problem 5-13.
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Problem 5-14.
5-6
mode
Median PX x xx
Y
Y
()ln .ln
d
§
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¹
¸
§
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P
V05 5
1
5.5. Exponential Distributions
Problem 5-15.
The total number of hurricanes for all years is
5-7
Problem 5-16.
(a) Average time between accidents
Problem 5-17.
For the exponential distribution, its parameter O can be computed from its mean as
O
1
25 04
..
Therefore, the following probabilities can be computed using the cdf of the exponential
distribution:
Problem 5-18.
For the exponential distribution, its parameter O can be computed from its mean as
day/2
O
5.6. Triangular Distributions
Problem 5-19.
5-8
Problem 5-20.
Pessimistic estimate = 90 days
Best estimate = 120 days
Optimistic estimate = 150 days
The area under the triangle has to be equal to 1. Therefore:
5.8. Raleigh Distributions
Problem 5-21.
0.025
0.03
0.035
Problem 5-22.
Parameter = 8.79100
22
S
P
S
D
x
Problem 5-23.
Mean = 10 ft
Problem 5-24.
COV=20/100=0.2, and k=7
a) P(S>120)
5-10
Problem 5-25.
a) Probability that it does not meet the demand?
b) Probability of having one 10-year storm in a year that exceeds the capacity of drainage
system?
Problem 5-26.
5.9. Statistical Probability Distributions
Problem 5-27.
Using Table A-2, the following probabilities based on the t-distribution can be computed:
2120 1746 0 05 0025 0049733...
..
Problem 5-28.
Problem 5-29.
Using Table A-2, the following quantities can be computed based on the t-distribution:
Problem 5-30.
Using Table A-3, the following probabilities based on the chi-square distribution can be computed:
Problem 5-31.
5-12
Problem 5-32.
Using Table A-3, the following quantities can be computed based on the chi-square distribution:
Problem 5-33.
Using Table A-4, the following quantities can be computed based on the F-distribution:
5.10. Extreme Value Distributions
Problem 5-35.
Using a mean of 25 ft and standard deviation of 5 ft, the parameters of extreme (largest) type I are
Problem 5-36.
Using a mean of 25 ft and standard deviation of 5 ft, the parameter k of extreme (largest) type II
needs to meet the following condition:
Problem 5-37.
Using a mean of 25 ft and standard deviation of 5 ft, the parameter w of extreme (largest) type III
is zero. The k parameter needs to meet the following condition:
5-13
5.13. Simulation and Probability Distributions
Problem 5-38.
The following table summarizes the random number generation and transformation:
X Z
0.2793 -0.585
0.8008 0.845
0.128 -1.136
0.6384 0.354
0.7554 0.692
0.0629 -1.531
0.3956 -0.265
0.6499 0.385
Problem 5-39.
Problem 5-40.
RandSeed
0.716359042 0.381514245 0.898080222 0.476997 0.208449
lognormvals
32.53084634 28.15121922 36.51891328 29.31057 25.87097
RandSeed
0.349708702 0.298259 0.00834 0.088305 0.735695
12.91004596 10.62574 0.251261 2.773488 39.91954
parameters:
S30
F^Ͳ1(randseed)/AKAexponentialvariates