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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam WFM-6204: Hydrologic Statistics Lecture-3: Probabilistic analysis: (Part-2) Akm Saiful Islam Institute of Water and Flood Management (IWFM) Bangladesh University of Engineering and Technology (BUET) December, 2006 WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Probability Distributions and Their Applications Continuous Distributions Normal distribution Lognormal distribution Gamma distribution Pearson Type III distribution Gumbel’s Extremal distribution WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Normal Distribution The probability that X is less than or equal to x when X can be evaluated N ( , 2 ) from x prob( X x) px ( x) (2 ) 2 1/ 2 e ( t )2 / 2 2 dt (4.9) The parameters (mean) and 2 (variance) are denoted as location and scale parameters, respectively. The normal distribution is a bell-shaped, continuous and symmetrical distribution (the coefficient of skew is zero). WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam If is held constant and 2 varied, the distribution changes as in Figure 4.2.1.1. Figure 4.2.1.1 Normal distributions with same mean and different variances WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam 2 If is held constant and varied, the distribution does not change scale but docs change location as in Figure 4.2.1.2. A common notation for indicating that a random variable is normally distributed with mean and variance 2 is N ( , 2 ) Figure 4.2.1.2 Normal distributions with same variance and different means WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam If a random variable is N ( , 2 ) and Y= a + bX 2 2 , the distribution of Y can be shown to be N (a b, b. ) This can be proven using the method of derived i 1,2, n distributions. Furthermore, if for X , are independently and normally distributed with mean i and variance , then Y a b X b X b X is normally distributed with i 2 i Y a i 1bi i n i 1bi2 i2 2 Y n 1 1 2 2 (4.7) and (4.8) Any linear function of independent normal random variables is also a normal random variable. n n WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Standard normal distribution The probability that X is less than or equal to x when X is N ( , ) can be evaluated from 2 x prob( X x) px ( x) (2 ) 2 1/ 2 e ( t )2 / 2 2 dt (4.9) WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam The equation (4.9) cannot be evaluated analytically so that approximate methods of integration arc required. If a tabulation of the integral was made, a separate table would be required for each value of and . By using the liner transformation Z ( X ) / , the random variable Z will be N(0,1). The random variable Z is said to be standardized (has 0 and 1 ) and N(0,1) is said to be the standard normal distribution. The standard normal distribution is given by z (4.10) p ( z) (2 ) e and the cumulative standard normal is given by z prob( Z z) PZ ( z) (2 ) 1 / 2 e t / 2 dt (4.11) 2 2 1 / 2 z2 / 2 Z 2 WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Figure 4.2.1.3 Standard normal distribution WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Figure 4.2.1.3 shows the standard normal distribution which along with the transformation Z ( X ) / contains all of the information shown in Figures 4.1 and 4.2. Both pZ (Z ) and PZ (z) are widely tabulated. Most tables utilize the symmetry of the normal distribution so that only positive values of Z are shown. Tables of PZ (z) may show prob( Z z ) or prob (0 Z z ) Care must be exercised when using normal probability tables to see what values are tabulated. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Example-1: As an example of using tables of the normal distribution consider a sample drawn from a N(15,25). What is the prob(15.6≤ X≤ 20.4)? [Hann] Solution: WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Example-2: What is the prob(10.5≤ X≤ 20.4) if X distributed N(15,25)? [Hann] Solution: WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Example-3: Assume the following data follows a normal distribution. Find the rain depth that would have a recurrence interval of 100 years. Year 2000 1999 1998 1997 1996 ….. Annual Rainfall (in) 43 44 38 31 47 ….. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Solution: Mean = 41.5, St. Dev = 6.7 in (given) x= Mean + Std.Dev * z x = 41.5 + z(6.7) P(z) = 1/T = 1/100 = 0.01 F(z) = 0.5 – P(z) = 0.49 From Interpolation using Tables E.4 Z = 2.236 X = 41.5 + (2.326 x 6.7) = 57.1 in WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Bi-Nominal Distribution PDF P ( x) Range 0 xn Mean np Variance np (1 p ) n! p x (1 p ) n x x! (n x)! WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Poisson Distribution PDF Range P( x) x e x! 0 x ... Mean Variance WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Normal Distribution PDF f ( x) 1 2 Range x Mean Variance 2 e ( x ) 2 / 2 2 WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Log-Normal Distribution ( y = ln x) PDF 1 ( y ) 2 / 2 2 f ( y) e x 2 Range y Mean Variance y 2 y WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Gamma Distribution PDF Range Mean Variance WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Gumbel Distribution PDF Range Mean Variance WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Extreme Value Type-1 Distribution PDF Range Mean Variance WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Properties of common distributions Log-Pearson III Distribution PDF Range Mean Variance WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam Assignment-1 The total annual runoff from a small drainage basin is determined to be approximately normal with a mean of 14.0 inch and a variance of 9.0 inch2. Determine the probability that the annual runoff form the basin will be less than 11.0 inch in all three of next the three consecutive years.