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CS145: Probability & Computing Lecture 9-a: Gaussian Distributions, Moment Generating Functions

CS145: Probability & Computing Lecture 9-a: Gaussian Distributions, Moment Generating Functions. Instructor: Eli Upfal Brown University Computer Science. Figure credits: Bertsekas & Tsitsiklis , Introduction to Probability , 2008 Pitman, Probability , 1999. CS145: Lecture 9-a Outline.

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CS145: Probability & Computing Lecture 9-a: Gaussian Distributions, Moment Generating Functions

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  1. CS145: Probability & ComputingLecture 9-a: Gaussian Distributions, Moment Generating Functions Instructor: Eli Upfal Brown University Computer Science Figure credits:Bertsekas & Tsitsiklis, Introduction to Probability, 2008Pitman, Probability, 1999

  2. CS145: Lecture 9-a Outline • Gaussian (Normal) Distributions • Moment Generating Function • Moments of the Normal Distribution

  3. Gaussian (Normal) Distributions

  4. Why the Normal Distribution? Binomial Distribution:

  5. Normal Binomial Approximation

  6. The Normal is a Proper Distribution

  7. Scaling a Gaussian Variable Any linear transformation of a Gaussian variable is Gaussian! • Mean and variance of linear functions: • Proof that PDF is Gaussian will come later …

  8. Standard Normal Random Variables

  9. Classic Computation of Normal CDF

  10. Modern Computation of Normal CDF

  11. CS145: Lecture 9-a Outline • Gaussian (Normal) Distributions • Moment Generating Function • Moments of the Normal Distribution

  12. Moment Generating Function

  13. Proof

  14. Generating Function of a Sum

  15. Examples

  16. CS145: Lecture 9-a Outline • Gaussian (Normal) Distributions • Moment Generating Function • Moments of the Normal Distribution

  17. Back to the Normal Distribution

  18. Moments of the Normal Distribution

  19. Expectation and Variance of N(,

  20. M.G.F Defines a Distribution

  21. Sum of Normal Random Variables

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