1. Moment Generating Functions

2. Continuous Distributions
The Uniform distribution from a to b

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

4. The Exponential distribution

5. Weibull distribution with parameters a and b.

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 a and 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 kth central moment of X
where m = 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. Examples
The Binomial distribution (parameters p, n)

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

36. Moments Moment generating functions
Summary
Moments
Moment generating functions

37. Moments of Random Variables
The moment generating function

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

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)