1. Use of moment generating functions

2. Definition Let X denote a random variable with probability density function f(x) if continuous (probability mass function p(x) if discrete) Then m X (t) = the moment generating function of X

3. The distribution of a random variable X is described by either 1.The density function f(x) if X continuous (probability mass function p(x) if X discrete), or 2.The cumulative distribution function F(x), or 3.The moment generating function m X (t)

4. Properties 1. m X (0) = 1 2. 3.

5. 4. Let X be a random variable with moment generating function m X (t). Let Y = bX + a Then m Y (t) = m bX + a (t) = E(e [bX + a]t ) = e at m X (bt) 5. Let X and Y be two independent random variables with moment generating function m X (t) and m Y (t). Then m X+Y (t) = m X (t) m Y (t)

6. 6. Let X and Y be two random variables with moment generating function m X (t) and m Y (t) and two distribution functions F X (x) and F Y (y) respectively. Let m X (t) = m Y (t) then F X (x) = F Y (x). This ensures that the distribution of a random variable can be identified by its moment generating function

7. M. G. F.’s - Continuous distributions

8. M. G. F.’s - Discrete distributions

9. Moment generating function of the gamma distribution where

10. using or

11. then

12. Moment generating function of the Standard Normal distribution where thus

13. We will use

14. Note: Also

15. Note: Also

16. Equating coefficients of t k, we get

17. Using of moment generating functions to find the distribution of functions of Random Variables

18. Example Suppose that X has a normal distribution with mean  and standard deviation . Find the distribution of Y = aX + b Solution: = the moment generating function of the normal distribution with mean a  + b and variance a 2  2.

19. Thus Z has a standard normal distribution. Special Case: the z transformation Thus Y = aX + b has a normal distribution with mean a  + b and variance a 2  2.

20. Example Suppose that X and Y are independent each having a normal distribution with means  X and  Y, standard deviations  X and  Y Find the distribution of S = X + Y Solution: Now

21. or = the moment generating function of the normal distribution with mean  X +  Y and variance Thus Y = X + Y has a normal distribution with mean  X +  Y and variance

22. Example Suppose that X and Y are independent each having a normal distribution with means  X and  Y, standard deviations  X and  Y Find the distribution of L = aX + bY Solution: Now

23. or = the moment generating function of the normal distribution with mean a  X + b  Y and variance Thus Y = aX + bY has a normal distribution with mean a  X + B  Y and variance

24. Special Case: Thus Y = X - Y has a normal distribution with mean  X -  Y and variance a = +1 and b = -1.

25. Example (Extension to n independent RV’s) Suppose that X 1, X 2, …, X n are independent each having a normal distribution with means  i, standard deviations  i (for i = 1, 2, …, n) Find the distribution of L = a 1 X 1 + a 1 X 2 + …+ a n X n Solution: Now (for i = 1, 2, …, n)

26. or = the moment generating function of the normal distribution with mean and variance Thus Y = a 1 X 1 + … + a n X n has a normal distribution with mean a 1  1 + …+ a n  n and variance

27. In this case X 1, X 2, …, X n is a sample from a normal distribution with mean , and standard deviations  and Special case:

28. Thus and variance has a normal distribution with mean

29. If x 1, x 2, …, x n is a sample from a normal distribution with mean , and standard deviations  then Summary and variance has a normal distribution with mean

30. Population Sampling distribution of

31. If x 1, x 2, …, x n is a sample from a distribution with mean , and standard deviations  then if n is large The Central Limit theorem and variance has a normal distribution with mean

32. We will use the following fact: Let m 1 (t), m 2 (t), … denote a sequence of moment generating functions corresponding to the sequence of distribution functions: F 1 (x), F 2 (x), … Let m(t) be a moment generating function corresponding to the distribution function F(x) then if Proof: (use moment generating functions) then

33. Let x 1, x 2, … denote a sequence of independent random variables coming from a distribution with moment generating function m(t) and distribution function F(x). Let S n = x 1 + x 2 + … + x n then

38. Is the moment generating function of the standard normal distribution Thus the limiting distribution of z is the standard normal distribution Q.E.D.