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 ( , 2.9), Chapter 3 ( )

12. Parametric vs Nonparametric
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. San Juan River Annual Streamflow 1906-1983
Histogram, bin width 150
Flow KAF
Number

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
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