Lecture 4
Discrete Probability Distribution (1)
Statistics for
Civil & Environmental Engineers
Bernoulli Distribution(1)
Description
there are only two possible outcomes, called a successor failure
the probability of occurrence of a success (or a failure) is constant
Bernoulli Distribution(2)
Bernoulli pmf
Population parameters
Mean=Variance=
Statistics for Civil & Environmental Engineers
Binomial Distribution(1)
Description
a series of Bernoulli trialsis made
the trials are conducted under the same conditions
the order in which the events in the trials occur is immaterial
Binomial pmf
Mean=
Variance=
Skewness coef.=
Binomial Distribution(2)
Population parameters
Parameter estimates
Statistics for Civil & Environmental Engineers
Binomial Distribution(3)
Comments
events must be independent
events must have only 2 possible outcomes (e.g., success or failure)
probability must be STABLE
for n=1, binomial reduces to Bernoulli distribution
Binomial Distribution Example
Statistics for Civil & Environmental Engineers
Negative Binomial Distribution(1)
Description
a series of Bernoulli trials that are continued until exactly r success occur when X trials are required
Negative Binomial pmf
Negative Binomial Distribution(2)
Population parameters
Comments
for r =1, the negative binomial distribution reduces to the geometric distribution
Mean=
Variance=
Distribution shapes
8 10 12 14 16 18 20 22 0.1
x p = 0.25
0 6 7 8 9 10 12 14 0.4
x r = 5
0.2
p = 0.75 r = 5 0.2
Statistics for Civil & Environmental Engineers
Negative Binomial Distribution Example
Geometric Distribution
Description
the probability that the first occurrence is at the X-th time
Geometric pmf
Population parameters
Mean=Variance=
Statistics for Civil & Environmental Engineers
Geometric Distribution
Comments
the probability distribution of the length of time between occurrences
Distribution shapes
0 1 2 3 4 5 6 7 8
1/2
x p= 3/4
0 1 2 3 4 5 6 7 8
1/2
x p= 1/2
The mean time interval between exceedances of a specified value
The reciprocal of the probability of exceedance
Return Period
Definition
Equation
Statistics for Civil & Environmental Engineers
Return Period Example
Hypergeometric Distribution(1)
Description
drawing a random sample of size nwithout replacement from a finite population of size mwith the elements of the
population divided into two groups with mpelements belonging to one group
Negative Binomial pmf
Statistics for Civil & Environmental Engineers
Hypergeometric Distribution(2)
Population parameters
Comments
if nis “small” compared to m, the binomial distribution is reasonable approximation for the hypergeometric distribution Mean=
Variance=
Distribution shapes
N=10; mp=4; n=4 N=10; mp=4; n=5
Hypergeometric Distribution Example
Statistics for Civil & Environmental Engineers
