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Chapter 6 Continued

Chapter 6 Continued. MORE Continuous Probability Distributions. Extreme Value Distributions. Used when interested in extreme events such as maximum peak discharge of a stream OR minimum daily flows. The extreme value of a set of random variables is also a r.v.

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Chapter 6 Continued

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  1. Chapter 6 Continued MORE Continuous Probability Distributions

  2. Extreme Value Distributions • Used when interested in extreme events such as maximum peak discharge of a stream OR minimum daily flows. • The extreme value of a set of random variables is also a r.v. • The probability distribution of an extreme value r.v. will in general depend on the sample size and parent distribution from which the sample was obtained.

  3. Extreme Value Distribution • Consider a random sample of size n consisting of x1,x2, ….xn. • Let Y be the largest of the sample values. • Let PY(y) be prob(Y≤y) and PXi(x) be prob(Xi≤x) with pY(y) and pXi(x) as the corresponding pdfs.

  4. Extreme Value Distribution • If the x’s are independently and identically distributed: • The probability distribution of the maximum of n independently and identically distributed r.v. depends on the sample size, n, and the parent distribution, PX(x) of the sample.

  5. Extreme Value Distributions • Frequently we don’t know the parent distribution, but if the sample size is large, use can be made of certain asymptotic results. • Three types of asymptotic distributions have been developed based on different (but not all) parent distributions.

  6. Type I Distributions • The parent distribution is unbounded in the direction of the desired extreme and all moments exist. • Example parent distributions • Extreme value largest-normal, exponential, gamma • Extreme value smallest-normal

  7. Type II Distributions • Parent distribution unbounded in the direction of the desired extreme and all moments do not exist. • Example parent distribution • Extreme value largest or smallest- Cauchy distribution

  8. Type III Distributions • The parent distribution is bounded in the direction of the desired extreme. • Example parent distributions: • Extreme value largest-beta distribution • Extreme value smallest-beta, lognormal, gamma, exponential

  9. Extreme Value Type I (Gumbel) • The type I asymptotic distribution for maximum (minimum) values is the limiting model as n approaches infinity for the distribution of the maximum (minimum) of n independent values from an initial distribution whose right (left) tail is unbounded and which is an exponential type. • The initial cumulative distribution approaches unity (zero) with increasing (decreasing) values of the r.v. as fast as the exponential distribution approaches unit • Examples of use: Distribution of daily and peak river flows.

  10. Extreme Value Type I • (-) applies for the maximum • (+) applies for the minimum • a and b are scale and location parameters • b is the mode of the distribution • The type I for maximum and minimum values are symmetrical with each other about the mode.

  11. Extreme Value Type I Parameters maximum minimum maximum minimum

  12. Transformed EV Type I • If you use the transformation: The type I extreme value pdf becomes

  13. Transformed EV I cdf maximum Double exponential minimum

  14. Method of moments estimators maximum minimum

  15. Extreme Value Type III Minimum (Weibull) • Arises when extreme is from a parent distribution that is limited in the direction of interest. • Example: Distribution of low stream flows.

  16. Weibull pdf and cdf

  17. Parameters of the Weibull

  18. Method of Moments Estimators Solve simultaneously for and Let

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