Shahid Lecture-13-MKAG1273

MAL1303: STATISTICAL
HYDROLOGY
Frequency Distribution
Dr. Shamsuddin Shahid
Associate Professor
Department of Hydraulics and Hydrology
Faculty of Civil Engineering
Room No.: M46-332; Phone: 07-5531624;
Email: sshahid@utm.my
11/23/2015 Shamsuddin Shahid, FKA, UTM
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 Discrete Distributions
 Binomial Distribution
 Poisson Distribution
 Continuous Distributions
 Normal Distribution
 Lognormal Distribution
 Gamma Distribution
 Exponential Distribution
 Gumbel Distribution
Different Types of Probability Distribution

Random variables can be two types:
1. Discrete random variables have a countable number of
outcomes. For example: Flood/No Flood, Rainy days in a year,
etc.
2. Continuous random variables have an infinite continuum of
possible values. Fro example: Rainfall, River Discharge, etc.
Random Variable: Types

 A probability function maps the
possible values of random variable
(x) against their respective
probabilities of occurrence, p(x)
 p(x) is a number from 0 to 1.0.
 The area under a probability
function is always 1.
Probability Function

A probability mass function (pmf)
is a function that gives the
probability that a discrete random
variable is exactly equal to some
value.
The probability mass function is
often the primary means of
defining a discrete probability
distribution.
Probability Mass Function (pmf)
x p(x)
1 p(x=1) = 1/6
2 p(x=2) = 1/6
3 p(x=3) = 1/6
4 p(x=4) = 1/6
5 p(x=5) = 1/6
6 p(x=6) = 1/6
1.0

Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF), or the distribution function,
describes the probability that a random variable with a given probability
distribution will be found at a value less than or equal to x.
x p(x)
1 p(x 1) = 1/6
2 p(x 2) = 2/6
3 p(x 3) = 3/6
4 p(x 4) = 4/6
5 p(x 5) = 5/6
6 p(x 6) = 6/6

1. What’s the probability of getting 2 or less?
2. What’s the probability of getting 5 or higher?
x p(x)
1 p(x 1) = 1/6
2 p(x 2) = 2/6
3 p(x 3) = 3/6
4 p(x 4) = 4/6
5 p(x 5) = 5/6
6 p(x 6) = 6/6

Which of the following are probability functions?
a. f(x)=0.2 for x=1,2,3,4,5
b. f(x)= (x-2)/4 for x=1,2,3,4
c. f(x)= (x2+x-5)/8 for x=2,3,4
Is the Function is a Probability Function
x p(x)
1 f(x=1) = 0.2
2 f(x=2) = 0.2
3 f(x=3) = 0.2
4 f(x=4) = 0.2
5 f(x=5) = 0.2
1.0
x p(x)
1 f(x=1) = -0.25
2 f(x=2) = 0.0
3 f(x=3) = 0.25
4 f(x=4) = 0.5
x p(x)
1 f(x=2) = 0.125
2 f(x=3) = 0.875
3 f(x=4) = 1.875
>1.0

Find the probability of storm in a give year:
Exactly 7 storms, p(x=7)= 0.1
At least 7 storms, p(x>=7) = (0.1+0.1) = 0.2
At most 6 storms, p(x<=6) = (0.5 + 0.3) = 0.8
x 5 6 7 8
p(x) 0.5 0.3 0.1 0.1
The number of storms occur in a year is represented
by a random variable x. From analysis of historical
data, it was found that the probability distribution for
x is:
Use of Probability
10 year data:
2000 6
2001 5
2002 6
2003 8
2004 7
2005 5
2006 6
2007 5
2008 5
2009 5

Let us consider a negative exponential function,
x
exf 
)(
110
0
0






xx
ee
The probability distribution of variable x is called Exponential Distribution.
This function integrates to 1:
Continuous Distribution Function

The probability that x is any
exact particular value (such as x
= 1.2) is 0. We can only assign
probabilities to possible ranges
of x. For example, The
probability of x between 1 and 2
is :
Probability Density Function (PDF)
23036801350
2)xP(1
12
2
1
2
1
...
eee
e
x
x







we can specify the “cumulative distribution function” (CDF), P(x≤A),
AAA
A
x
A
x
eeeeee 
 110
0
0
0.8650.135-1
-12)P(x 2

 
e
Probability of random variable
less than or equal to 2,P(x≤2),

 
0.135
0.865-1
-1-1
2)(x-12)P(x
2





e
Probability of random variable greater than or equal to 2,P(x2),

Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
Rainfall
(mm)
49.1 48.5 26.7 50.9 31.8 44.7 78.5 28.5 65.8 66.2 73.6 102.2 78 55.2 45.3
The probability density function of an exponential distribution is
Find the probability the hourly annual maximum rainfall
exceeds a threshold of 38mm, P(X > 38).

 Continuous Distributions
 Normal Distribution
 Lognormal Distribution
 Gamma Distribution
 Exponential Distribution
 Extreme value distribution
 Gumbel Distribution
 -
 -
Different Types of Probability Distribution

Example-1

Example-2

Example-3

• One of the simplest continuous distributions in all of statistics
is the continuous uniform distribution.
• This distribution is characterized by a density function that is
“flat,” and thus the probability is uniform in a closed interval.
• Applications of the continuous uniform distribution are not
wide.
• The density function of the continuous uniform random
variable X on the interval [A, B] is
Continuous Uniform Distribution

• The density function forms a rectangle with base B−A and
constant height 1/B−A.
• As a result, the uniform distribution is often called the
rectangular distribution.

Suppose that a flood in an area never last for more than 4 days. Both long
and short floods occur quite often. In fact, it can be assumed that the
length X of a flood has a uniform distribution on the interval [0, 4].
(a) What is the probability density function?
(b) What is the probability that any flood lasts at least 3 days?
ANSWER:
(a) The appropriate density function for the uniformly distributed random
variable X in this situation is
(b) P[X  3] =
4
1
4
1
4
3
 dx

The mean and variance of the uniform distribution are:
Mean:
Variance:
2
BA
µ


12
2
2 )AB(
σ



Assume the following rainfall data follows a normal distribution.
Find the rain depth that would have a recurrence interval of 100
years.
Year Annual Rainfall (in)
2000 43
1999 44
1998 38
1997 31
1996 47
….. …..
Mean = 41.5, St. Dev = 6.7 in
Normal Distribution

Solution:
Z = (X − µ)/σ
X = µ + (Z * σ)
x = 41.5 + z(6.7)
Return period, T = 100
Probability of occurrence in a year, 1/T = 1/100 = 0.01
Z = 2.326
X = 41.5 + (2.326 x 6.7) = 57.1 in
Normal Distribution

Frequency Analyses
Primary application of flood
frequency analyses is to predict the
possible flood magnitude over a
certain time period or to estimate
the frequency with which floods of
a certain magnitude may occur.
• Time distribution of flood
• Estimation of the magnitude of
flood
• Estimation of return period

• A 100-year flood
does not
necessarily occur
only once every
100 years, nor
will it
necessarily occur
only once during
a 100 year
period.
• There is a equal
chance for a
flood of this
magnitude to
occur in any year
or even multiple
times in a single
year.

Frequency Analysis
 Rank the (n) data (Pi) in a descending order, the highest value first
and the lowest value last.
 Attach a serial rank number, r to each value (Pi) with r = 1 for the
highest value (Pi) and r = n for the lowest value (Pn)
 Calculate the frequency of exceedance F (P>Pi) as:
California r / n
Hazen (r – 0.5)/n
Weibull r / (n+1)
Gringorten (r – 0.44) / (n + 0.12)

Flood Return Period

• The method of moments equates sample moments to
parameter estimates.
• The moments are measured are mean, variance, skewness
and kurtosis.
• When moment methods are available, they have the
advantage of simplicity.
• The disadvantage is that they are often not available and
they do not have the desirable optimality properties of other
methods.
Using Moments

There are various methods, both numerical and
graphical, to test goodness of fit:
1. Probability plots
2. Statistical tests
Test The Goodness of Fit

Shahid Lecture-13-MKAG1273

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