09 Probability and Statistics W08 1

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PROBABILITY AND

STATISTICS

87-323: HydrologyWinter 2009


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87-323 HydrologyUniversity of Windsor 2


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Statistics To extract the essential information from a set of data, reducing

a large set of numbers to a small set of numbers

make generalizations about populations using informationobtained from random sampling

population

Based on mathematical principles that describe the random

variation of a set of observations or a process

Focus on the observations themselves rather than on thephysical processes which produced them



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Sample and sample space vs Population

amp e s a ran om co ec on su se opopulation

ven

Probability



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Statistical Measures

Mean first moment ofvalues about the ori in

Variance or Standard

Skewness Third moment



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Effect of changes in and Cs on the

ro a y ens y unc on

Cs > 0Cs = 0 Cs < 0



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Probabilistic Treatment of

Probability is the chance that an even will occur whenan observation of the random variable is made

a continuous random variable

probability distribution function specifies the chancethat an observation xof the variable will fall in aspecified range ofX (lets say annual precipitation)

they will have a non-zero probability



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Typical Questions -

that P(R45 ?

P(35R45)?

Assuming annual

precipitation is an ,calculate the

will be two successiveyears of precipitationless than 35



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Return Period and Probability

Hydrology - involves Stochastic Processes

Partly deterministic and Partly random

Hydrologic processes or observations are described

Discrete an Ran om Varia es

2 or 5 year rainfall

100- ear flood

An annual maximum (or any other independent event)even has a return period of T years, if its magnitude is

, , . 1-F =(1/T) the probability that the event equaled or

exceeded in an sin le ear



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Flow-Duration Curve

Plot of magnitude vs.ercent of time the

magnitude is equaledor exceeded.



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Probability functions



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Probability Distributions for

The objective of a discrete analysis is most often to assignpro a t es to t e num er o occurrences o an event

Whereas, the objective of a continuous analysis is mostoften to determine the probability of the magnitude ofan event, and vice versa.

Normal distribution

Log-normal distribution

xponen a s r u on Gamma/Pearson Type III distribution



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( )

=

xexf

x 2

2

121

)(

Normal Distribution

duezF

z u

= 22

1)(

xz

Log-normal DistributionIf the log of a random variable (RV) is normallydistributed, the RV is considered to have Log-normaldistribution

Calculate the statistical measures and probabilitydistribution functions for the logarithmic values of the

is a log-Pearson distribution with Cs =0



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Normal

s r u on

P(z


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Fitting A Probability Distribution

Method of moments Easier, suitable for

Method of Maximum likelihood

esting of the Goodness of fit

Comparison of theoretical and sample valuesof the relative frequency or the cumulativefunction



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to relate the magnitude of events to the

requency o occurrence roug e use oprobability distributions

independent and identically distributed

Hydrologic system producing them is considered,

A r hProbability paperMathematical Approach



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Plot between the value of random variablevs probability of its occurrence

Probability scale depends on the

distribution bein usedDistributions normal or Gumbel Extreme



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Commonly used model for normal, log-normal, Pearson Type III and log-Pearsonype III distributions

X=X+KSRandom Variable Mean K reflects the

probability of occurrence

o va ue

87-323 Hydrology

University of Windsor 20


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ai

Genericaand bare constants depends on Prob Distribution

1+

=ban

Pi

Weibull

1+

=n

iPi

Hazen iPi

5.0=

87-323 Hydrology

University of Windsor 21

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09 Probability and Statistics W08 1