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Moment Generating Functions

Moment Generating Functions. Contents. Review of Continuous Distribution Functions. Continuous Distributions. The Uniform distribution from a to b. The Normal distribution (mean m , standard deviation s ). The Exponential distribution. The Gamma distribution.

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Moment Generating Functions

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  1. Moment Generating Functions

  2. Contents • Review of Continuous Distribution Functions

  3. Continuous Distributions The Uniform distribution from a to b

  4. The Normal distribution (mean m, standard deviation s)

  5. The Exponential distribution

  6. The Gamma distribution Then X is said to have a Gamma distribution with parameters aand l. Let the continuous random variable X have density function:

  7. Moment Generating function of a Random Variable X

  8. Moment Generating function of a Random Variable X • The Binomial distribution (parameters p, n) Examples

  9. Moment Generating function of a Random Variable X • The Poisson distribution (parameter l) The moment generating function of X , mX(t) is:

  10. Moment Generating function of a Random Variable X • The Exponential distribution (parameter l) The moment generating function of X , mX(t) is:

  11. Moment Generating function of a Random Variable X • The Standard Normal distribution (m = 0, s = 1) The moment generating function of X , mX(t) is:

  12. Moment Generating function of a Random Variable X We will now use the fact that We have completed the square This is 1

  13. Moment Generating function of a Random Variable X • The Gamma distribution (parameters a, l) The moment generating function of X , mX(t) is:

  14. Moment Generating function of a Random Variable X We use the fact Equal to 1

  15. Properties of Moment Generating Functions Note: the moment generating functions of the following distributions satisfy the property mX(0) = 1 • mX(0) = 1

  16. Properties of Moment Generating Functions We use the expansion of the exponential function:

  17. Properties of Moment Generating Functions Now

  18. Properties ofMoment Generating Functions

  19. Properties ofMoment Generating Functions Property 3 is very useful in determining the moments of a random variable X. Examples

  20. Properties of Moment Generating Functions

  21. Properties of Moment Generating Functions

  22. Properties of Moment Generating Functions To find the moments we set t = 0.

  23. Properties of Moment Generating Functions

  24. Properties of Moment Generating Functions

  25. Properties of Moment Generating Functions

  26. Properties of Moment Generating Functions The moments for the exponential distribution can be calculated in an alternative way. This is note by expanding mX(t) in powers of t and equating the coefficients of tkto the coefficients in:

  27. Properties of Moment Generating Functions Equating the coefficients of tkwe get:

  28. The moments for the standard normal distribution We use the expansion of eu.

  29. The moments for the standard normal distribution We now equate the coefficients tk in:

  30. Properties of Moment Generating Functions For even 2k: If k is odd: mk= 0.

  31. The log of Moment Generating Functions Let lX(t) = lnmX(t) = the log of the moment generating function

  32. The log of Moment Generating Functions

  33. The log of Moment Generating Functions Thus lX(t) = lnmX(t) is very useful for calculating the mean and variance of a random variable

  34. The log of Moment Generating Functions • The Binomial distribution (parameters p, n) Examples

  35. The log of Moment Generating Functions

  36. The log of Moment Generating Functions • The Poisson distribution (parameter l)

  37. The log of Moment Generating Functions • The Exponential distribution (parameter l)

  38. The log of Moment Generating Functions

  39. The log of Moment Generating Functions • The Standard Normal distribution (m = 0, s = 1)

  40. Summary

  41. Summary

  42. Expectation of functions of Random Variables X is discrete X is continuous

  43. Moments of Random Variables The kth moment of X

  44. Moments of Random Variables The 1th moment of X

  45. Moments of Random Variables wherem = m1 = E(X) = the first moment of X . The kthcentralmoment of X

  46. Rules for expectation Rules:

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