shahid lecture-13-mkag1273

MAL1303: STATISTICAL HYDROLOGY

Frequency Distribution

Dr. Shamsuddin ShahidAssociate Professor

Department of Hydraulics and HydrologyFaculty of Civil Engineering

Room No.: M46-332; Phone: 07-5531624; Email: sshahid@utm.my

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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

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Random variables can be two types:

1. Discrete random variables have a countable number ofoutcomes. For example: Flood/No Flood, Rainy days in a year,etc.

2. Continuous random variables have an infinite continuum ofpossible values. Fro example: Rainfall, River Discharge, etc.

Random Variable: Types

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A probability function maps thepossible values of random variable(x) against their respectiveprobabilities of occurrence, p(x)

p(x) is a number from 0 to 1.0.

The area under a probabilityfunction is always 1.

Probability Function

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A probability mass function (pmf)is a function that gives theprobability that a discrete randomvariable is exactly equal to somevalue.

The probability mass function isoften the primary means ofdefining a discrete probabilitydistribution.

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

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Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF), or the distribution function,describes the probability that a random variable with a given probabilitydistribution 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

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1. What’s the probability of getting 2 or less?

2. What’s the probability of getting 5 or higher?

Cumulative Distribution Function (CDF)

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

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Which of the following are probability functions?a. f(x)=0.2 for x=1,2,3,4,5b. f(x)= (x-2)/4 for x=1,2,3,4c. 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) = -0.252 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

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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 representedby a random variable x. From analysis of historicaldata, it was found that the probability distribution forx is:

Use of Probability

10 year data:2000 62001 52002 62003 82004 72005 52006 62007 52008 52009 5

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Let us consider a negative exponential function,

xexf )(

11000

xx ee

The probability distribution of variable x is called Exponential Distribution. This function integrates to 1:

Continuous Distribution Function

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The probability that x is anyexact particular value (such as x= 1.2) is 0. We can only assignprobabilities to possible rangesof x. For example, Theprobability of x between 1 and 2is :

Probability Density Function (PDF)

23036801350

2)xP(1

122

1

2

1

...

eee

e

x

x

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we can specify the “cumulative distribution function” (CDF), P(x≤A),

AAAA

xA

x eeeeee 110

00

Cumulative Distribution Function (CDF)

0.865 0.135-1 -1 2)P(x 2

e

Probability of random variableless than or equal to 2,P(x≤2),

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Cumulative Distribution Function (CDF)

0.135

0.865 -1 -1-1

2)(x-1 2)P(x 2

e

Probability of random variable greater than or equal to 2,P(x2),

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Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009Rainfall (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).

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Continuous Distributions Normal Distribution Lognormal Distribution Gamma Distribution Exponential Distribution Extreme value distribution Gumbel Distribution - -

Different Types of Probability Distribution

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Example-1

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Example-2

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Example-3

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Example-3

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• One of the simplest continuous distributions in all of statisticsis 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 notwide.

• The density function of the continuous uniform randomvariable X on the interval [A, B] is

Continuous Uniform Distribution

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• The density function forms a rectangle with base B−A andconstant height 1/B−A.

• As a result, the uniform distribution is often called therectangular distribution.

Continuous Uniform Distribution

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Continuous Uniform Distribution

Suppose that a flood in an area never last for more than 4 days. Both longand short floods occur quite often. In fact, it can be assumed that thelength 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] =41

414

3

dx

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Continuous Uniform Distribution

The mean and variance of the uniform distribution are:

Mean:

Variance:

2BAµ

12

22 )AB(σ

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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 431999 441998 381997 311996 47….. …..Mean = 41.5, St. Dev = 6.7 in

Normal Distribution

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

Normal Distribution

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Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009Rainfall (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).

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Frequency Analyses

Primary application of floodfrequency analyses is to predict thepossible flood magnitude over acertain time period or to estimatethe frequency with which floods ofa certain magnitude may occur.

• Time distribution of flood• Estimation of the magnitude of

flood• Estimation of return period

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• A 100-year flooddoes notnecessarily occuronly once every100 years, norwill itnecessarily occuronly once duringa 100 yearperiod.

• There is a equalchance for aflood of thismagnitude tooccur in any yearor even multipletimes in a singleyear.

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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 thehighest value (Pi) and r = n for the lowest value (Pn)

Calculate the frequency of exceedance F (P>Pi) as:California r / nHazen (r – 0.5)/nWeibull r / (n+1)Gringorten (r – 0.44) / (n + 0.12)

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Flood Return Period

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• The method of moments equates sample moments toparameter estimates.

• The moments are measured are mean, variance, skewnessand kurtosis.

• When moment methods are available, they have theadvantage of simplicity.

• The disadvantage is that they are often not available andthey do not have the desirable optimality properties of othermethods.

Using Moments

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There are various methods, both numerical andgraphical, to test goodness of fit:

1. Probability plots2. Statistical tests

Test The Goodness of Fit

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