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WFM 6204: Hydrologic Statistics © Dr. 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 Islam

WFM 6204: Hydrologic Statistics © Dr. 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

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Table: Area under standardized normal distribution

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Central Limit theorem

If Sn is the sum of n independently and identically distributed random variables Xi each having a mean and variance then in the limit as n approaches infinitely, the distribution of Sn approaches a normal distribution with mean n and variance n

2

2

WFM 6204: Hydrologic Statistics © Dr. 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.

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Normal Approximations

Bi-nomial Distribution If X is a binomial random variable with parameters n1 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=n1+n2.

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

)1(/)(/)( pnpnpXXZ

WFM 6204: Hydrologic Statistics © Dr. 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<X≤8).

WFM 6204: Hydrologic Statistics © Dr. 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 =1+2 .. 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.

2/1/)(/)( XXZ

WFM 6204: Hydrologic Statistics © Dr. 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<X≤8).

WFM 6204: Hydrologic Statistics © Dr. 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.

xX exp )( 0,0 x

X Xt

X edtexp0

1)( 0x

/1)( XE 2/1)var( X

X/1ˆ

WFM 6204: Hydrologic Statistics © Dr. 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.

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Area (acres) No. depressions Area (acres) No. depressions

0-½ 106 4-4½ 4

½-1 36 4½-5 5

1-1½ 18 5-5½ 2

1½-2 9 5½-6 6

2-2½ 12 6-6½ 3

2½-3 2 6½-7 1

3-3½ 5 7-7½ 1

3½-4 1

WFM 6204: Hydrologic Statistics © Dr. 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.

,)(

)()(

1

r

ettf

tr

0 0r

)(t

WFM 6204: Hydrologic Statistics © Dr. 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.

dyeyr

tFt yr

0

1

)(

1)( ).,(

)(

1tr

r

/rm 22 / r

WFM 6204: Hydrologic Statistics © Dr. 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)

)(

)()(

)(1

r

ettf

trr

t

0 t

/rm

WFM 6204: Hydrologic Statistics © Dr. 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) = 200 + 3/1 0-3 = 3200 hr.

8.0)3,4,3()3(

1)2004500(10,3

)3(

1)4500( 3

F

/10 3

WFM 6204: Hydrologic Statistics © Dr. 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.

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Log-Normal DistributionThe 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:

2

2

1

22

1)(

y

yy

y

eyf

y

XY ln

WFM 6204: Hydrologic Statistics © Dr. 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 .

22

2

1

22

2

1

2 2

11

2

1)()(

y

y

y

y y

y

y

y

exx

edx

dyyfxf

y 2y

y 2y

WFM 6204: Hydrologic Statistics © Dr. 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 Xs is:

The coefficient of skew of the Xs is: Thus the lognormal distribution is skewed to the right;

the skewness increasing with increasing values of .

XSC xv /

1ln2

12

2

vC

XY

)1ln( 22 vy CS

XY lnn

yy

i1

22

2

n

ynyS

i

y

)2/exp()( 2yyXE 1)(

22 yeXVar x

12

yeCv

33 vv CC

vC

y 2y

and

WFM 6204: Hydrologic Statistics © Dr. 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

WFM 6204: Hydrologic Statistics © Dr. 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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