Functions of Random Variables

1. Functions of Random Variables

2. Methods for determining the distribution of functions of Random Variables • Distribution function method • Moment generating function method • 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 1 Let X have a normal distribution with mean 0, and variance 1. (standard normal distribution) Let W = X2. Find the distribution of W.

5. First step Find the distribution function of W G(w) = P[W ≤ w] = P[ X2≤ w] where

6. Second step Find the density function of W g(w) = G'(w).

7. Thus if X has a standard Normal distribution then W = X2 has density This distribution is the Gamma distribution with a = ½ and l = ½. This distribution is also the c2 distribution with n = 1 degree of freedom.

8. Example 2 Suppose that X and Y are independent random variables each having an exponential distribution with parameter l(mean 1/l) Let W = X + Y. Find the distribution of W.

9. First step Find the distribution function of W = X + Y G(w) = P[W ≤ w] = P[ X + Y ≤ w]

14. Second step Find the density function of W g(w) = G'(w).

15. Hence if X and Y are independent random variables each having an exponential distribution with parameter lthen W has density This distribution can be recognized to be the Gamma distribution with parameters a = 2 and l.

16. Example: Student’s t distribution Let Z and U be two independent random variables with: • Z having a Standard Normal distribution • and • U having a c2 distribution with n degrees of freedom Find the distribution of

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

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

19. Therefore:

20. Illustration of limits t > 0 t > 0 U z z U

21. Now: and:

22. Using:

23. If then Using the fundamental theorem of calculus: then

24. Hence Using or

25. Hence and

26. or where

27. Student’s t distribution where

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

29. t distribution standard normal distribution

30. Distribution of the Max and Min Statistics

31. Let x1, x2, … , xndenote a sample of size n from the density f(x). Let M = max(xi) then determine the distribution of M. Repeat this computation for m = min(xi) Assume that the density is the uniform density from 0 to q.

32. Hence and the distribution function

33. Finding the distribution function of M.

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

35. Finding the distribution function of m.

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

37. 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.

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

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

40. Theorem Let X denote a random variable with probability density function f(x) and U = h(X). Assume that h(x) is either strictly increasing (or decreasing) then the probability density of U is: The Transformation Method

41. Proof Use the distribution function method. Step 1 Find the distribution function, G(u) Step 2 Differentiate G (u ) to find the probability density function g(u)

42. hence

43. or

44. Suppose that X has a Normal distribution with mean mand variance s2. Find the distribution of U = h(x) = eX. Solution: Example

45. hence This distribution is called the log-normal distribution

46. log-normal distribution

47. Theorem Let x1, x2,…, xn denote random variables with joint probability density function f(x1, x2,…, xn ) Let u1= h1(x1, x2,…, xn). The Transfomation Method(many variables) u2= h2(x1, x2,…, xn). ⁞ un= hn(x1, x2,…, xn). define an invertible transformation from the x’s to the u’s

48. Then the joint probability density function of u1, u2,…, un is given by: where Jacobian of the transformation

49. Suppose that x1, x2 are independent with density functions f1 (x1) and f2(x2) Find the distribution of u1= x1+ x2 Example u2= x1 - x2 Solving for x1 and x2 we get the inverse transformation

50. The Jacobian of the transformation