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Moment Generating Functions

Moment Generating Functions. The Uniform distribution from a to b. Continuous Distributions. The Normal distribution (mean m , standard deviation s ). The Exponential distribution. Weibull distribution with parameters a and b . The Weibull density, f ( x ). ( a = 0.9, b = 2).

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Moment Generating Functions

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  1. Moment Generating Functions

  2. The Uniform distribution from a to b Continuous Distributions

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

  4. The Exponential distribution

  5. Weibull distribution with parameters a andb.

  6. The Weibull density, f(x) (a= 0.9, b= 2) (a= 0.7, b= 2) (a= 0.5, b= 2)

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

  8. Expectation of functions of Random Variables

  9. X is discrete X is continuous

  10. Moments of Random Variables

  11. The kth moment of X.

  12. the kthcentralmoment of X wherem = m1 = E(X) = the first moment of X .

  13. Rules for expectation

  14. Rules:

  15. Moment generating functions

  16. Moment Generating function of a R.V. X

  17. The Binomial distribution (parameters p, n) Examples

  18. The Poisson distribution (parameter l) The moment generating function of X , mX(t) is:

  19. The Exponential distribution (parameter l) The moment generating function of X , mX(t) is:

  20. The Standard Normal distribution (m = 0, s = 1) The moment generating function of X , mX(t) is:

  21. We will now use the fact that We have completed the square This is 1

  22. The Gamma distribution (parameters a, l) The moment generating function of X , mX(t) is:

  23. We use the fact Equal to 1

  24. Properties of Moment Generating Functions

  25. mX(0) = 1 Note: the moment generating functions of the following distributions satisfy the property mX(0) = 1

  26. We use the expansion of the exponential function:

  27. Now

  28. Property 3 is very useful in determining the moments of a random variable X. Examples

  29. To find the moments we set t = 0.

  30. 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 tk to the coefficients in: Equating the coefficients of tk we get:

  31. The moments for the standard normal distribution We use the expansion of eu. We now equate the coefficients tk in:

  32. For even 2k: If k is odd: mk= 0.

  33. Summary Moments Moment generating functions

  34. Moments of Random Variables The moment generating function

  35. The Binomial distribution (parameters p, n) Examples • The Poisson distribution (parameter l)

  36. The Exponential distribution (parameter l) • The Standard Normal distribution (m = 0, s = 1)

  37. The Gamma distribution (parameters a, l) • The Chi-square distribution (degrees of freedom n) (a = n/2, l = 1/2)

  38. Properties of Moment Generating Functions • mX(0) = 1

  39. The log of Moment Generating Functions Let lX (t) = ln mX(t) = the log of the moment generating function

  40. Thus lX (t) = ln mX(t) is very useful for calculating the mean and variance of a random variable

  41. The Binomial distribution (parameters p, n) Examples

  42. The Poisson distribution (parameter l)

  43. The Exponential distribution (parameter l)

  44. The Standard Normal distribution (m = 0, s = 1)

  45. The Gamma distribution (parameters a, l)

  46. The Chi-square distribution (degrees of freedom n)

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