1. 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, p , all except joint moment generating function p134.
3.4 Associated Random Variables and Probabilities, p (Skip moment generating function of derived variables to end of chapter)
Benjamin and Cornell
Derived Distributions, p

2. 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

3. Discrete Variables - Joint Probability Mass Function

4. 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

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

6. Conditional Probability Mass Function

7. Marginal Probability Mass Function

8. Marginal Probability Mass Function

9. Extending Conditional Probability

10. Joint Probability Distributions of Continuous Variables
( corrected)

11. Generalized to arbitrary region

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

13. Conditional Distribution

14. Marginal Distribution

15. Expectation and moments of multivariate random variables

16. Conditional Expectation
Discrete
Continuous

17. Conditional Expectation
Table 3.3.1
Table 3.3.2

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

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

20. 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

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