Probability and Statistics Reference Chapter 2

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Probability and StatisticsReference Chapter 2

• Marty Wanielista
• 407.823.4144
• http//stormwater.ucf.edu
• http//people.cecs.ucf.edu/wanielista
• http//classes.cecs.ucf.edu/CWR4101/wanielista

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Why Use Probability and Statistics in Hydrology?

• Can not predict a hydrologic event with certainty
• Examples of rainfall and streamflow
• What is the volume of rainfall over 6 hours?
• What is the rainfall depth in the month of July?
• What is the stream flow in a river?
• Infiltration rates? Soils are heterogeneous.
• Groundwater flow and levels?
• What is the design rainfall volume for a highway?

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Most recent interest related to hurricanes spins
off new vocabularies

• Variable in time (stochastic)
• FEMA (federal emergency management agency)
• FIS (flood insurance studies)
• BFE (base flood elevation) maps
• DFIRM (digital flood insurance rate maps)

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Basic Concepts

• Independence
• 4 hours for meteorological
• Time for runoff is dependent on watershed
• Peak Streamflow example
• Conditional Probability Pr(X) x/n condition
  
• Probability of a rain day in July? In January?
• Geographic, time of day, inter-event dry period?
• Graphical Presentations
• Empirical Frequency distributions and histograms

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Basic Concepts

• Statistical Descriptions
• Central Tendency 1st Moment Mean, Median, Mode
• Measure of Spread 2nd Moment Variance, SD, SE,
  Cv
• Skewness Third Moment, G
• Exceedence such as gt, and gt
• Return Period, inverse of Exceedence
• Theoretical Distributions because we simply do
  not have sufficient samples.

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Population Parameters vs. Sample Statistics
_
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Population Parameters vs. Sample Statistics,
Continued
1/2

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Population Parameters vs. Sample Statistics,
Continued
Note G Cs
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Return Period (Tr)

• Given Tr 100 yr. , what is the probability that
  a given event will occur in any one year?
• Pr 1/(Tr) 1/100 0.01

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Return Period, Tr 1/Pr(X ? x)

• X some event or variable, e.g., annual peak
  flood
• Specific Design Return Period, e.g., 10 yr
• x basis, e.g., what is the value in cfs
• Pr(X ? x) the probability that the event will
  exceed or equal a given basis in a single time
  period, e.g., the probability the annual peak
  discharge will exceed or equal to x cfs.

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Binomial Distribution

• Given p Pr(X ? x)
• The probability, that X ? x will occur (x) times
  out of n events, is given by the Binomial
  Distribution

(
)
n x
px (1- p)n-x
Pr(x)
n x
Recall, ( ) n?/x?(n-x)?
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• What is the probability that an event, X, will
  not occur in n time periods?

(
)
Pr(x0n,p) (1-p)n (p)0
n 0
And Pr (0 n,p) (1-p)n
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Specified Return Period Storm will occur at
least once.

• Given Risk, R 1- (1 - p)n
• Let p 1/Tr
• R 1 - (1- 1/Tr)n
• Tr 1/1- (1-R)1/n

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Weather Systems

• Frontal Movements
• Thunder storms (convective)
• Heat Islands
• Cyclonic Storms
• Orographic Effects

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Yearly volumes greatest in the panhandle Small
volumes in the Keys
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Probability Distributions

• Determine which theoretical distribution best
  fits an empirical one
• Must develop an empirical one based on a limited
  sample size
• Then compare to theoretical ones.

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For the empirical data, determine the empirical
probabilities or use a Plot Position (m) Given
a number of data points (n)

• Weibull m/(n1)
• California m/n
• Foster (2m-1)/2n
• Exceedence (m-1)/n
• Must rank order the data ( usually from lowest to
  highest value)

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Discrete Flow Data (Max yearly)
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Weibull Plotting Position(Discrete)

• Steps
• Order the data
• Estimate an empirical probability
• Calculate an exceedence probability
• Calculate return period

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Ordered from Low to High
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Other Plotting Positions(Discrete)
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Theoretical Distributions in Hydrology

• Normal
• Log Normal
• Gumbel G1.14
• Log Pearson Type 3
• Skewed to the right distributions (G is ) are
  the most common in hydrology

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Exceedence Probability for the Gumbel Distribution

• where

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Example Problem 2.6 (page 36) If flood flows on
a large watershed have an average value of 1,200
cms with a varianece of 62,500 (cms)2, what is
the probability that a flood will be equal to or
exceed 2,000 cms using the Gumbel distribution?
Solution

• ?x 1,200 cms
• s2 62,500 cms2 ? s 250 cms
• x 2,000

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Regression Analysis

• How well one measured variable can be used to
  predict another variable.
• Criteria Minimize the sum of the square of the
  difference between the measured and predicted
  variables.
• Correlation Coefficient R, measures the goodness
  of fit 0lt R lt1

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The Coefficient of Determinant, R2
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Summary

• Calculate Probabilities and Statistics
• Empirical Distributions lead to Theoretical ones
• Exceedence and Return Period
• Risk
• Regression and Correlation