1. Functions of Random Variables

2. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating function method 3.Transformation method

3. Distribution function method Let X, Y, Z …. have joint density f(x,y,z, …) Let W = h( X, Y, Z, …) First step Find the distribution function of W G(w) = P[W ≤ w] = P[h( X, Y, Z, …) ≤ w] Second step Find the density function of W g(w) = G'(w).

4. Example: Student’s t distribution Let Z and U be two independent random variables with: 1. Z having a Standard Normal distribution and 2. U having a  2 distribution with degrees of freedom Find the distribution of

5. The density of Z is: The density of U is:

6. Therefore the joint density of Z and U is: The distribution function of T is:

7. Then where

8. Student’s t distribution where

9. Student – W.W. Gosset Worked for a distillery Not allowed to publish Published under the pseudonym “Student

10. t distribution standard normal distribution

11. Distribution of the Max and Min Statistics

12. Let x 1, x 2, …, x n denote a sample of size n from the density f(x). Let M = max(x i ) then determine the distribution of M. Repeat this computation for m = min(x i ) Assume that the density is the uniform density from 0 to .

13. Hence and the distribution function

14. Finding the distribution function of M.

15. Differentiating we find the density function of M. f(x)f(x)g(t)g(t)

16. Finding the distribution function of m.

17. Differentiating we find the density function of m. f(x)f(x)g(t)g(t)

18. The probability integral transformation This transformation allows one to convert observations that come from a uniform distribution from 0 to 1 to observations that come from an arbitrary distribution. Let U denote an observation having a uniform distribution from 0 to 1.

19. Find the distribution of X. Let Let f(x) denote an arbitrary density function and F(x) its corresponding cumulative distribution function. Hence.

20. has density f(x). Thus if U has a uniform distribution from 0 to 1. Then U

21. Use of moment generating functions

22. 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

23. 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)

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

25. 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)

26. 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

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

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

29. Moment generating function of the gamma distribution where

30. using or

31. then

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

33. We will use

34. Note: Also

35. Note: Also

36. Equating coefficients of t k, we get

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

38. 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.

39. 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.

40. 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

41. 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

42. 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

43. 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

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

45. 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)

46. 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

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

48. Thus and variance has a normal distribution with mean

49. 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

50. Population Sampling distribution of

51. 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

52. 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

53. 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

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