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Chapter 6

Chapter 6. Still more continuous distributions. 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. Weibull pdf and cdf. Parameters of the Weibull .

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Chapter 6

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  1. Chapter 6 Still more continuous distributions

  2. 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.

  3. Weibull pdf and cdf

  4. Parameters of the Weibull

  5. Maximum Likelihood Estimators Solve simultaneously for and Let

  6. Weibull distribution

  7. 3 Parameter Weibull • When lower bound of parent distribution is not zero, a 3rd parameter e, known as the displacement parameter must be included. Can use this transformation for easier calculation of probabilities.

  8. Mean and Variance of 3 parameter Weibull

  9. Estimation procedure for a, b and e • Calculate the skew, g. • Use Table 6.2 to find g, a, and A(a) and B(a). • Calculate mean, m and standard deviation, s.

  10. Beta Distribution • Has both upper and lower bound. • Generally defined over interval 0 to 1. • Can be transformed to any interval from a to b. • If the bounds of the distribution are unknown it becomes a 4 parameter distribution (a, b, a, b)

  11. PDF of Beta Distribution Beta function; tabulated. Related to the gamma function

  12. Mean and Variance of Beta Distribution (MOM estimators)

  13. Distributions of Sample Statistics • Since sample statistics (mean, variance, skew, kurtosis) are functions of random variables, they themselves are random variables. • Statistical tests depend on the distribution of “test statistics” which are sample statistics. • 3 common distributions of sample statistics are chi-square, t and F distributions.

  14. Chi-Square Distribution • Z is the standardized normally distributed random variable where: • We can define Y as: • Y has chi-square distribution with n degrees of freedom. • Chi-square is a special case of the gamma distribution with l = ½ and h a multiple of ½.

  15. Pdf for the chi-square distribution One parameter distribution with n= 2h (degrees of freedom)

  16. Mean and Variance and Skew of Chi-Square

  17. Cumulative chi-square distribution Tabulated in appendix A.14 for various values of a and n.

  18. The t distribution • If Y is a standardized normal variate and U is a chi-square variate with n degrees of freedom AND Y and U are independent then has a t distribution with n degrees of freedom. AS n gets large the t-distribution approaches the standard normal distribution.

  19. Pdf and cumulative t distribution Tabulated in Appendix A. 13 for various values of a and n.

  20. Mean and Variance of t distribution E(T) = 0 For n > 2

  21. Uses for t distribution • Sampling distribution for the mean from a normal distribution with unknown variance. Y has a standard normal distribution U has a chi-square distribution.

  22. F distribution • If U is a chi-square variate with g = m degrees of freedom and V is a chi-square variate with g = n degrees of freedom and U and V are independent then X = (U/m)/(V/n) has an F distribution with g1=m and g2=n degrees of freedom (numerator and denominator degrees of freedom, respectively)

  23. The F distribution pdf

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