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Multiple Random Variables

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.

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Multiple Random Variables

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  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, 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

  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

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

  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 (3.3.11 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.

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