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

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

33. 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:

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

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

36. Summary Moments Moment generating functions

37. Moments of Random Variables The moment generating function

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

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

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

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

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

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

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

45. The Poisson distribution (parameter l)

46. The Exponential distribution (parameter l)

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

48. The Gamma distribution (parameters a, l)

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