Chapter 4-2 Continuous Random Variables

1. Chapter 4-2Continuous Random Variables 主講人:虞台文

2. Content • Functions of Single Continuous Random Variable • Jointly Distributed Random Variables • Independence of Random Variables • Distribution of Sums • Distributions of Multiplications and Quotients • Conditional Densities • Multivariate Distributions • Multidimensional Changes of Variables

3. Chapter 4-2Continuous Random Variables Functions of Single Continuous Random Variable

4. The Problem 已知 =?

5. Example 11 已知 =?

6. Example 11

7. Example 11

8. Example 12

9. Example 12 請熟記! 標準常態之平方為一個自由度的卡方

10. Example 13 0 as 0  y <16 0 as 0  y  4

11. Example 13 0 as 0  z  2 0 as 0  z <4

12. fX(x) 1 x Example 14

13. fX(x) 1 x Example 14 How to generate exponentially distributed random numbers using a computer?

14. fX(x) 1 x Example 14

15. or g Theorem 1 Let g be a differentiable monotone function on an interval I, and let g(I) denote its range. Let X be a continuous r. v. with pdf fX such that fX(x) = 0 for x  I. Then, Y = g(X) has pdf fY given by

16. or g Theorem 1 Let g be a differentiable monotone function on an interval I, and let g(I) denote its range. Let X be a continuous r. v. with pdf fX such that fX(x) = 0 for x  I. Then, Y = g(X) has pdf fY given by

17. Y = g(X) y X Theorem 1 Y = g (X) and Pf) Case 1: positive g1(y)

18. Y = g(X) y X Theorem 1 Y = g (X) and Pf) Case 2: negative g1(y)

19. fX(x) 1 x 1 Y = g (X) and Example 15 Redo Example 14 using Theorem 1.

20. Y = g (X) and Example 16

21. Y = g (X) and Example 16

22. Example 17

23. Example 17

24. Example 17

25. Example 17

26. Example 17  < 1 : DFR  = 1 : CFR  > 1 : IFR

27. Example 18 Let X be a continuous random variable. Define Y to be the cdf of X, i.e., Y = FX(X). Find fY(y).

28. Random Number Generation • The method to generate a random number X such that it possesses a particular distribution by a computer: • Let Y = FX(X). • Find • Generate a random variable by a computer in interval (0, 1). Let y be such a random number. • Computing , we obtain the desired random number x.

29. Example 19 How to generate a random variable X by a computer such that X ~ Exp()? • Let Y = FX(X) = 1  eX. So, Y ~ U(0, 1). • Assume U(0, 1) can be generated by a computer. • By letting X = 1ln(1Y), we then have X ~ Exp().

30. Chapter 4-2Continuous Random Variables Jointly Distributed Random Variables

31. Definition Joint Distribution Functions The joint (cumulative) distribution function (jcdf) of random variables X and Y is defined by: FX,Y(x, y) = P(X x, Y  y), < x < , < y < . (x, y)

32. Properties of a jcdf (x2, y2) (x1, y1)

33. d c a b (b, d) (a, d) Properties of a jcdf (b, c) (a, c)

34. Definition Marginal Distribution Functions Given the jpdf F(x, y) of random variables X, Y. The marginal distribution functions of X and Y are defined respectively by

35. Definition Joint Probability Density Functions A joint probability density function (jpdf) of continuous random variableX, Y is a nonnegative function fX,Y(x,y) such that

36. Properties of a Jpdf fX(u) fY(v)

37. Properties of a Jpdf Marginal Probability Density Functions (see next page) fX(u) fY(v)

38. Marginal Probability Density Functions

39. Example 20

40. Example 20

41. Example 20

42. Example 20

43. Example 20

44. Example 20

45. Example 20

46. Y 1 X 1 Example 20 0 1 0 0 0 0

47. Y 1 X 1 Example 20 0 1 0 (x, y) 0 0 0

48. Y 1 X 1 Example 20 0 1 (x, y) 0 (x, y) 0 0 0

49. Y 1 X 1 Example 20 0 1 (x, y) (x, y) 0 (x, y) 0 0 0

50. Example 20