# Probability Functions-3 ```Department of Civil Engineering
Statistics
for
Construction
Random Variables and Probability distributions
Part 3
Prepared by:
Dr. Bahram Abedinianagerabi
Outline
• Triangular Distribution
• Poisson Distribution
• Normal Distribution
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Triangular distribution (Cont’d)
• Probability density function (PDF):
μ = (a + b + c)/3
Var(x) = (a2+b2+c2 - ab - ac - bc)/18
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Triangular distribution (Cont’d)
• Example 6:
Duration of a construction project is the followings:
 Optimistic = 160 days
 Pessimistic = 185 days
 Most likely = 170 days
Use triangular distribution to answer the following questions:
a)
The mean and standard deviation:
b) What is the probability that the project duration is at least 172 days?
c)
What is 90 percentile completion time?
Solution
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Poisson distribution
• The Poisson distribution is a discrete probability distribution that
expresses the probability of a given number of events occurring in a fixed
interval of time or space if these events occur with a known constant mean
rate and independently of the time since the last event.
• The Poisson distribution can be applied to systems with a large number of
possible events, each of which is rare.
• A discrete random variable X is said to have a Poisson distribution with
parameter λ &gt; 0, if, for k = 0, 1, 2, ..., the probability mass function of X.
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Poisson distribution (Cont’d)
• The Poisson random variable satisfies the following conditions:
1.
The number of successes in two disjoint time intervals is independent.
2.
The probability of a success during a small time interval is proportional to the entire
length of the time interval.
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Poisson distribution (Cont’d)
• Examples
 Number of deaths by horse kicking in the Prussian army
 Birth defects and genetic mutations
 Rare diseases (like Leukemia, but not AIDS because it is infectious and so not
independent) - especially in legal cases
 Car accidents
 Traffic flow and ideal gap distance
 Number of typing errors on a page
 Hairs found in McDonald's hamburgers
 Spread of an endangered animal in Africa
 Failure of a machine in one month
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Poisson distribution (Cont’d)
e = 2.71828
• The positive real number λ is equal to the expected value of X and also to its
variance.
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Poisson distribution (Cont’d)
• Example 7:
In DFW area, snowing occurs once every 500 days on average.
a)
Calculate the probability of k = 3 snowing days in a 500 day interval,
assuming the Poisson model is appropriate.
b) Calculate the probability having less than 5 snowing days in a 500 day
interval.
c)
Calculate the probability having equal or more than 3 snowing days in a 500
day interval.
Solution
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Poisson distribution (Cont’d)
• Example 8:
Vehicles pass through a junction on a busy road at an average rate of 300 per
hour.
a)
Find the probability that none passes in a given minute.
b) What is the expected number passing in two minutes?
c)
Find the probability that this expected number actually pass through in a
given two-minute period.
Solution
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Normal distribution
• Normal distribution, also known as the Gaussian distribution, is a
probability distribution that is symmetric about the mean, showing that
data near the mean are more frequent in occurrence than data far from
the mean.
• In graph form, normal distribution will appear as a bell curve.
Parameters: the mean, , and the standard deviation, .
mean  = central tendency
standard deviation  = measure of spread.
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Normal distribution
• Notation:
• Probability density function:
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Normal distribution (Cont’d)
• Some characteristics of a normal distribution are:
 Continuous data which has a bell shaped histogram.
 The MEAN is in the middle.
 The distribution is symmetrical.
A lower
mean
Total Area = 1
A higher
mean
x
x
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Normal distribution (Cont’d)
• Some characteristics of a normal distribution are:
1 Std Dev either side of
mean = 68%
2 Std Dev either side of
mean = 95%
3 Std Dev either side of
mean = 99%
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Normal distribution (Cont’d)
• Distributions with different spreads have different STANDARD
DEVIATIONS
A smaller Std Dev.
A larger Std Dev.
Graph
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Normal distribution (Cont’d)
• Standard Normal Distribution Properties:
 Total area under curve = 1
 The mean = 0
 The standard deviation = 1
 The curve is symmetrical
 P(a &lt; Z &lt; b) = shaded area
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Normal distribution (Cont’d)
• How to solve problems related to Normal Distributions
• The question can be determining the probability of X less or
greater than a number
Steps:
2) Convert X to ‘z’
4) Look up ‘z’ in the tables to get the probability
Graphical displays of Normal Distribution
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Normal distribution (Cont’d)
• Example 9:
The heights of the same variety of pine tree are also normally distributed. The
mean height is μ=33mm and the standard deviation is 3.
Which normal distribution below best summarizes the data?
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Normal distribution (Cont’d)
• Example 10:
Assume that the duration of a construction activity has normal distribution with
the mean of 10 days and standard deviation of 1.5 days.
a)
What is the probability which the activity takes 10 days or less?
b) What is the probability which the activity takes 12 days or more?
Z table
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Normal distribution (Cont’d)
• Example 11:
The final exam scores in a statistics class were normally distributed with a mean
of 63 and a standard deviation of 5.
a)
Find the probability that a randomly selected student scored more than 65
on the exam.
b) Find the probability that a randomly selected student scored less than 85.
c)
Find the 90th percentile (that is, find the score k that has 90% of the scores
below k and 10% of the scores above k).
d) Find the 70th percentile (that is, find the score k such that 70% of scores are
below k and 30% of the scores are above k).
Z table
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Normal distribution (Cont’d)
• Example 12:
A personal computer is used for office work at home, research, communication,
personal finances, education, entertainment, social networking, and a myriad of
other things. Suppose that the average number of hours a household personal
computer is used for entertainment is two hours per day. Assume the times for
entertainment are normally distributed and the standard deviation for the times
is half an hour.
a)
Find the probability that a household personal computer is used for
entertainment between 1.8 and 2.75 hours per day.
b) Find the maximum number of hours per day that the bottom quartile of
households uses a personal computer for entertainment.
Z table 2
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Normal distribution (Cont’d)
• Example 13:
A construction company pays its employees an average wage of \$3.25 an hour
with a standard deviation of 60 cents. If the wages are approximately normally
distributed, determine:
a)
the proportion of the workers getting wages between \$2.75 and \$3.69 an
hour;
b) the minimum wage of the highest 5%.
Z table
Z table 2
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