Multiple Random Variables Describe the representation of randomness in different variables that occur simultaneously or are related. Text and Readings Kottegoda and Rosso 3.3 Multiple Random Variables, p118-139, all except joint moment generating function p134. 3.4 Associated Random Variables and Probabilities, p139-154. (Skip moment generating function of derived variables to end of chapter) Benjamin and Cornell Derived Distributions, p100-123

Conditional and Joint Probability Definition Bayes Rule If D and E are independent Partition of domain into non overlaping sets D1 D2 D3 E Larger form of Bayes Rule

Discrete Variables - Joint Probability Mass Function

Joint Probability Mass Function X=0 0.2910 0.0600 0.0000 X=1 0.0400 0.3580 0.0100 X=2 0.0250 0.1135 0.0300 X=3 0.0005 0.0015 0.0505

Conditional Probability Mass Function Rescale so that PX|Y(x|yj) adds to 1

Conditional Probability Mass Function

Marginal Probability Mass Function

Marginal Probability Mass Function

Extending Conditional Probability

Joint Probability Distributions of Continuous Variables (3.3.11 corrected)

Generalized to arbitrary region

Conditional and joint density functions, analogous to discrete variables Conditional density function Marginal density function If X and Y are independent

Conditional Distribution

Marginal Distribution

Expectation and moments of multivariate random variables

Conditional Expectation Discrete Continuous

Conditional Expectation Table 3.3.1 Table 3.3.2

Derived Distributions (Benjamin and Cornell, p100-123) From Benjamin and Cornell (1970, p107)

General Derived Distribution Pr[Y≤y]=Pr[X takes on any value x such that g(x)≤y] From Benjamin and Cornell (1970, p111)

The Monte Carlo Simulation Approach Streamflow and other hydrologic inputs are random (resulting from lack of knowledge and unknowability of boundary conditions and inputs) System behavior is complex Can be represented by a simulation model Analytic derivation of probability distribution of system output is intractable Inputs generated from a Monte Carlo simulation model designed to capture the essential statistical structure of the input variables Monte Carlo simulations solve the derived distribution problem to allow numerical determination of probability distributions of output variables

From Bras, R. L. and I. Rodriguez-Iturbe, (1985), Random Functions and Hydrology, Addison-Wesley, Reading, MA, 559 p.