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: