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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam WFM-6204: Hydrologic Statistics Akm Saiful Islam Lecture-5: Probabilistic analysis: (Part-3) December, 2006 Institute of Water and Flood Management (IWFM) Bangladesh University of Engineering and Technology (BUET) WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Probability Distributions and Their Applications Continuous Distributions Normal distribution Exponential distribution Gamma distribution Lognormal distribution Extreme value distribution Extreme value type-I: Gumbel distribution Extreme value type-III (minimum): Weibull distribution Pearson Type III distribution

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Table: Area under standardized normal distribution

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Central Limit theorem If S n is the sum of n independently and identically distributed random variables X i each having a mean and variance then in the limit as n approaches infinitely, the distribution of S n approaches a normal distribution with mean n and variance n

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Constructing normal curves for data Frequently the histogram of a set observed data suggests that the data may be approximated by a normal distribution. One way to investigate the goodness of this approximation is by superimposing a normal curve on the frequency histogram and then visually compare the two distributions. Statistical procedures for testing the hypothesis that a set of data can be approximated by a normal (or any other) distribution.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Normal Approximations Bi-nomial Distribution If X is a binomial random variable with parameters n 1 and p and Y is a binomial random variable with parameters n2 and p the Z=X+Y is a binomial random variable with parameters n=n 1 +n 2. Central Limit theorem would indicate that the normal distribution approximates the binomial distribution if n is large. Thus as n gets large the distribution of Approaches a N(0,1). This is sometimes knows as the DeMoivre-Laplace limit theorem

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-1: [Haan, 1979] X is a binomial random variable with n=25 and p=0.3. Compare the binomial and normal approximation to the binomial for evaluating the prob(5

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Normal Approximations Poission Distribution The sum of two Poissions random variables with parameters 1 and 2 is also a Poissions random variable with parameter = Extending this to the sum of a large number of Poission random variables, the Central Limit Theorem indicates that for large, the Poission may be approximated by a normal distribution. In this case the distribution of approaches a N(0,1). Since the Poission is the limiting form of the binomial and the binomial can be approximated by the normal, it is no surprise that the Poission can also be approximated by the normal.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Assignment-2:[due next week] X is a Poission random variable with =np where n=25 and p=0.3. Compare the Poission and normal approximation to the Poission for evaluating the prob(5

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Exponential Distribution The exponential density function is given by and the cumulative exponential by The mean and variance of the exponential distribution are The exponential distribution is positively skewed with the skewness coefficient of 2. Both the method of moments and the maximum likelihood estimation give the parameter. The exponential distribution is a special case of the gamma distribution.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-2: [Haan, 1979] Haan and Johnson (1967) studied the physical characteristics of depressions in north-central Iowa. The data tabulated below shows the number of depressions falling into various classes based on the surface area of the depression. Plot a relative frequency histogram of the data. Superimpose on the histogram the best fitting exponential distribution. Estimate the probability that a depression selected at random will have an area greater than 2.25 acres.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Area (acres)No. depressionsArea (acres)No. depressions 0-½1064-4½4 ½-1364½ ½185-5½2 1½-295½ ½126-6½3 2½-326½ ½57-7½1 3½-41

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Gamma Distribution The gamma distribution failure probability density obeys the equation, (4.6.1) where parameter r need not be an integer. The two parameters are the shape parameter r and the scale parameter. The shape of the distribution depends significantly upon the value of r. It has also an impact on the hazard rate. In the special case that r is an integer, the Erlangian distribution is recovered; in the special case that = 0.5 and r = 0.5, where is the number of degrees of freedom, the gamma distribution becomes the chi-square distribution.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam The cumulative failure probability F(t) is: (4.6.2) The mean and variance of the gamma distribution are: and The gamma distribution is especially appropriate for systems subjected to an environment of repetitive, random shocks generated according to the Poisson distribution; thus the failure probability depends upon how many shocks the device has suffered, i.e., its age. As another application, if the mean rate of wear of a device is a constant, but the rate of wear is subject to random variations, then the gamma function should be used.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam For some devices, such as those for which corrosion of metals is important, it may be appropriate to modify the two-parameter gamma distribution by introducing a time delay before the onset of failures begins. Then equation (4.6.1) is modified to read as: (4.6.3) In such a case, the mean of the distribution becomes: (4.6.4)

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-3:[MacCormick,1981] Suppose that a device subjected to repetitive random shocks satisfies a gamma distribution with parameters r = 3 and hr, and that no failures can occur until 200 hour have passed. Estimate (a) the probability of failure after the device has operated for t = 4500 hour and (b) its mean time to failure. (MacCormick, 1981, p. 37) Solution:In this problem, the time displacement is = 200 hour. Integrating equation (4.6.3) from 0 to t and using equation (4.6.2) gives the cumulative probability: Using equation (4.6.4) gives mean time to failure (MTTF) = /1 0-3 = 3200 hr.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-4: [Haan, 1979] The annual water yield for Cave Creek (near Fort Spring, Kentucky) is shown in the following table. Estimate the parameters of the gamma distribution for this data using both the method of moments and the method of maximum likelihood. Assuming the data follows a gamma distribution, estimate the probability of an annual water yield exceeding 20.0 inch.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Log-Normal Distribution The lognormal distribution (sometimes spelled out as the logarithmic normal distribution) of a random variable is one for which the logarithm of follows a normal or Gaussian distribution. Denote, then Y has a normal or Gaussian distribution given by:

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Derived distribution: Since,, the distribution of can be found as: (4.13.2) Note that equation (4.13.1) gives the distribution of Y as a normal distribution with mean and variance. Equation (4.13.2) gives the distribution of X as the lognormal distribution with parameters and.

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Estimation of parameters (, ) of lognormal distribution: Note:,, Chow (1954) Method: (1) (2) (3) The mean and variance of the lognormal distribution are: The coefficient of variation of the X s is: The coefficient of skew of the X s is: Thus the lognormal distribution is skewed to the right; the skewness increasing with increasing values of. and

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-5: [Haan, 1979, p. 78] Use the lognormal distribution and calculate the expected relative frequency for the third class interval on the data in table 5.1

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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-6: [Haan, 1979] Assume the data of table 5.1 follow the lognormal distribution. Calculate the magnitude of the 100-year peak flood.

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