1. Moments of Random Variables
The moment generating function

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

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

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

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

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

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

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

9. The Poisson distribution (parameter l)

10. The Exponential distribution (parameter l)

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

12. The Gamma distribution (parameters a, l)

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

16. Jointly distributed Random variables
Multivariate distributions

17. Discrete Random Variables

18. The joint probability function;
p(x,y) = P[X = x, Y = y]

19. Marginal distributions
Conditional distributions

20. Continuous Random Variables

21. Definition: Two random variable are said to have joint probability density function f(x,y) if

22. Marginal distributions
Conditional distributions

23. Independence

24. Definition:
Two random variables X and Y are defined to be independent if
if X and Y are discrete
if X and Y are continuous
Thus in the case of independence
marginal distributions ≡ conditional distributions

25. The Multiplicative Rule for densities
if X and Y are discrete
if X and Y are continuous

26. Proof:
This follows from the definition for conditional densities:
Hence
and
The same is true for continuous random variables.

27. Example:
Suppose that a rectangle is constructed by first choosing its length, X and then choosing its width Y.
Its length X is selected form an exponential distribution with mean m = 1/l = 5. Once the length has been chosen its width, Y, is selected from a uniform distribution form 0 to half its length.
Find the probability that its area A = XY is less than 4.

28. Solution:

29. xy = 4
y = x/2

31. This part can be
evaluated
This part may require Numerical evaluation