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

Explore Gaussian (Normal) Distributions, Moment Generating Functions, and the properties of the Normal Distribution in this informative CS145 lecture. Understand why the Normal Distribution is essential, how it approximates the Binomial Distribution, and more.

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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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