1. Moment Generating Functions 1/33

2. Contents Review of Continuous Distribution Functions 2/33

3. Continuous Distributions The Uniform distribution from a to b

4. The Normal distribution (mean , standard deviation  )

5. The Exponential distribution

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

7. Moment Generating function of a Random Variable X

8. Examples 1.The Binomial distribution (parameters p, n)

9. The moment generating function of X, mX(t) is: 2.The Poisson distribution (parameter ) Moment Generating function of a Random Variable X

10. The moment generating function of X, mX(t) is: 3.The Exponential distribution (parameter ) Moment Generating function of a Random Variable X

11. The moment generating function of X, mX(t) is: 4.The Standard Normal distribution (  = 0,  = 1) Moment Generating function of a Random Variable X

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

13. The moment generating function of X, mX(t) is: 4.The Gamma distribution (parameters , ) Moment Generating function of a Random Variable X

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

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

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

17. Now Properties of Moment Generating Functions

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

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

26. 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: Properties of Moment Generating Functions

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

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

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

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

31. The log of Moment Generating Functions Let l X (t) = ln m X (t) = the log of the moment generating function

32. The log of Moment Generating Functions

33. Thus l X (t) = ln m X (t) is very useful for calculating the mean and variance of a random variable

34. The log of Moment Generating Functions Examples 1.The Binomial distribution (parameters p, n)

35. The log of Moment Generating Functions

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

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

39. 4.The Standard Normal distribution (  = 0,  = 1) The log of Moment Generating Functions

40. Summary

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

43. Moments of Random Variables The k th moment of X

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

45. The k th central moment of X where  =  1 = E(X) = the first moment of X. Moments of Random Variables

46. Rules for expectation Rules: