Probability Distributions

1. Probability Distributions • Learn the common parametric probability distributions that are used to represent the randomness of physical phenomena • Learn the physical mechanisms that underlie common probability distributions • Apply physical knowledge about the mechanisms that lead to randomness in a process to select an appropriate probability distribution to fit • Distribution selection is more than selecting the best fit empirical curve Reading Kottegoda and Rosso Chapter 4.

2. Kottegoda and Rosso page 201

3. Kottegoda and Rosso page 228

4. From http://www.weibull.com/

5. Carl Friedrich Gauß, immortalized

6. Normal Distribution Standardized Cumulative Use Normal tables or normcdf, norminv

7. Linear combinations of normally distributed variables Regenerative Central Limit Theorem for n→ The normal distribution is the natural distribution to characterize any process where the randomness is due to the additive effect of many independent inputs, without domination by one specific input.

8. Log Normal Distribution X is log normally distributed if Y=ln(X) is normally distributed Derived distribution Parameters y, y, pertain to the ln of the variable.

9. Lognormal and Normal distribution relationships used for simulation Moments Coefficient of Variation Estimation of parameters from moments

10. Log normal distribution and central limit theorem Parameters , , pertain to the ln of the variable, the Y’s. The log normal distribution is the natural distribution to characterize any process where the randomness is due to the multiplicative effect of many independent inputs, without domination by one specific input. (If one Xi can be 0 it makes the product 0, dominating the resulting distribution)

11. Nonparametric Density Estimation • Learn the kernel method for nonparametric density estimation • Learn the difference between fixed kernel window width and variable kernel window width estimates • Learn some of the approaches to kernel window width estimation • Reference • Least squares cross validation Reading Silverman Chapter 1, Chapter 2 (2.1-2.6, 2.9), Chapter 3 (3.1-3.4)

12. Parametric vs Nonparametric • Parametric • Assume data from a known distribution (e.g. Gamma, Normal) and estimate parameters • Nonparametric • Assume that the data has a probability density function but not of a specific known form • Let data speak for themselves • Exploratory data analysis

13. Which method conveys the information best to you ? Cumulative Density Probability Density Equation Probability Plot

14. Number Flow KAF San Juan River Annual Streamflow 1906-1983 Histogram, bin width 150

15. San Juan River Annual Streamflow 1906-1983 Density Flow KAF

16. Kernel Density Estimator

17. Kernel Density Estimate (KDE) • Place “kernels” at each data point • Sum up the kernels • Width of kernel determines level of smoothing • Determining how to choose the width of the kernel is an important topic! Narrow kernel Medium kernel Wide kernel

18. KDE window width sensitivity

19. Variable Kernel

20. San Juan River Annual Streamflow 1906-1983, Variable Kernel Density Estimate Density Flow KAF

21. Some Common Kernels

22. Density estimate with different kernels with the same bandwidth

23. Least Squares Cross Validation Density Mo(h) Flow KAF Window width h