1. Review of Probability Theory
CWR 6536
Stochastic Subsurface Hydrololgy

2. Random Variable (r.v.)
A variable (x) which takes on values at random, and may be thought of as a function of the outcomes of some random experiment.
The r.v. maps sample space of experiment onto the real line
The probability with which different values are taken by the r.v. is defined by the cumulative distribution function, F(x), or the probability density function, f(x).

3. Examples
Discontinuous r.v. - die tossing experiment

4. Examples
Categorical r.v. – An observation, s(a), that can take on any of a finite number of mutually exclusive, exhaustive states (sk) , e.g. soil type, land use, landscape position
An indicator random variable can be defined
The frequency of occurrence of a state f (sk) can be determined as the arithmetic average of n indicator data (i(a,sk) )where:
The joint frequency of two states sk and vk is

5. Frequency Table for Categorical Data
Soil type
Frequency
Land Use
Sand
20%
Forest
14%
Silt
33%
Pasture
21%
Loam
25%
Meadow
65%
Clay
22%
The probability distribution of categorical data is completely described
by a frequency table

6. Probability Density Function (pdf)
The function f(x) is a pdf for the continuous random variable x, defined over the set of real numbers R, if

7. Cumulative Distribution Function (cdf)
The cdf of a continuous r.v. x with a pdf f(x) is given by:
Properties of the cdf:

8. Examples: Continuous r.v.
uniform distribution,
exponential distribution,
gaussian distribution
log-normal distribution

9. Moments of a Random Variable
The pdf (and cdf) summarize all knowledge of the r.v., however we almost never really know this much about actual natural phenomena.
Moments of a r.v. provide a more aggregated description of its behavior which is often easier to estimate from field data than pdf or cdf

10. The First Moment
The expected value (or first moment, or population mean, or ensemble mean) of a r.v. is defined as the sum of all the values a r.v. may take, each weighted by the probability with which the value is taken
This quantity is a single valued, deterministic summary of the r.v.

11. Properties of the Expectation Operator
If the pdf, f(x) is even (i.e. f(x)=f(-x)), then the expected value is equal to zero
The expectation is a linear operator
The expected value of a function of the r.v.

12. Higher Order Moments n=1 E[x]=mean (measure of central tendency)
n=2 E[x2]= mean square
n=3 E[x3]= mean cube

13. Central Moments
n=1
n= variance
n= skewness
n= kurtosis
It can be shown that the full (infinite) set of moments completely exhausts the statistical information concerning the r.v., and thus the pdf can be constructed from the full set of moments

14. Joint Probability Distributions
The joint cdf for two random variables, x and y, is
where

15. Marginal cdfs and pdfs
cdfs
pdfs
conditional pdfs

16. Moments of two random variables

17. Statistical Independence vs Uncorrelation
Two random variables are statistically independent if
Two random variables are uncorrelated if
Two random variables are orthogonal if

18. Statistical Independence vs Uncorrelation
If two r.v. are independent then they are uncorrelated (but not vice versa)
If x and y are independent random variables then g(x) and h(y) are also independent random variables (this is not generally true if x and y are merely uncorrelated)
Correlation measures linear relatedness only
An exception is jointly normal r.v.s where uncorrelation
is equivalent to independence