WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam WFM-6204: Hydrologic Statistics Akm Saiful Islam Lecture-3: Probabilistic analysis: (Part-2) 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
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Probability Distributions and Their Applications Continuous Distributions Normal distribution Lognormal distribution Gamma distribution Pearson Type III distribution Gumbels Extremal distribution
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Normal Distribution The probability that X is less than or equal to x when X can be evaluated from The parameters (mean) and (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). (4.9)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam If is held constant and varied, the distribution changes as in Figure Figure Normal distributions with same mean and different variances
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam If is held constant and varied, the distribution does not change scale but docs change location as in Figure A common notation for indicating that a random variable is normally distributed with mean and variance is N Figure Normal distributions with same variance and different means
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam If a random variable is N and Y= a + bX, the distribution of Y can be shown to be. This can be proven using the method of derived distributions. Furthermore, if for, are independently and normally distributed with mean and variance, then is normally distributed with (4.7) and (4.8) Any linear function of independent normal random variables is also a normal random variable.
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Standard normal distribution The probability that X is less than or equal to x when X is can be evaluated from (4.9)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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, the random variable Z will be N(0,1). The random variable Z is said to be standardized (has and ) and N(0,1) is said to be the standard normal distribution. The standard normal distribution is given by (4.10) and the cumulative standard normal is given by (4.11)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Figure Standard normal distribution
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Figure shows the standard normal distribution which along with the transformation contains all of the information shown in Figures 4.1 and 4.2. Both and are widely tabulated. Most tables utilize the symmetry of the normal distribution so that only positive values of Z are shown. Tables of may show or prob Care must be exercised when using normal probability tables to see what values are tabulated.
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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 IslamDr. 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 IslamDr. 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. YearAnnual Rainfall (in) ….. …..
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Solution: Mean = 41.5, St. Dev = 6.7 in (given) x= Mean + Std.Dev * z x = 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 = X = (2.326 x 6.7) = 57.1 in
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Bi-Nominal Distribution PDF Range Mean Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Poisson Distribution PDF Range Mean Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Normal Distribution PDF Range Mean Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Log-Normal Distribution PDF Range Mean Variance ( y = ln x)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Gamma Distribution PDF Range Mean Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Gumbel Distribution PDF Range Mean Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Extreme Value Type-1 Distribution PDF Range Mean Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Properties of common distributions Log-Pearson III Distribution PDF Range Mean Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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 inch 2. 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.