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Chapter 5 Expectations

Chapter 5 Expectations

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Chapter 5 Expectations

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  1. Chapter 5 Expectations 主講人:虞台文

  2. Content • Introduction • Expectation of a Function of a Random Variable • Expectation of Functions of Multiple Random Variables • Important Properties of Expectation • Conditional Expectations • Moment Generating Functions • Inequalities • The Weak Law of Large Numbers and Central Limit Theorems

  3. Chapter 5 Expectations Introduction

  4. 有夢最美

  5. 有夢最美

  6. Definition  Expectation The expectation (mean), E[X] or X, of a random variable X is defined by:

  7. Definition  Expectation The expectation (mean), E[X] or X, of a random variable X is defined by: provided that the relevant sum or integral is absolutely convergent, i.e.,

  8. 有些隨機變數不存在期望值。 若存在則為一常數。 Definition  Expectation The expectation (mean), E[X] or X, of a random variable X is defined by: provided that the relevant sum or integral is absolutely convergent, i.e.,

  9. Example 1 Let X denote #defectives in the experiment.

  10. Example 2

  11. Example 3  驗證此為一正確之pdf

  12. Example 3

  13. Chapter 5 Expectations Expectation of a Function of a Random Variable

  14. The Expectation of Y=g(X)

  15. The Expectation of Y=g(X)

  16. Example 4

  17. Example 5

  18. 某些g(X)吾人特感興趣 第k次中央動差 第k次動差 第ㄧ次動差謂之均數(mean) 第二次中央動差謂之變異數(variance) Moments

  19. 均數、變異數與標準差 X:為標準差

  20. X ~ B(n, p) E[X]=?Var[X]=? Example 6

  21. X ~ B(n, p) E[X]=?Var[X]=? Example 6

  22. X ~ B(n, p) E[X]=?Var[X]=? Example 6

  23. X ~ Exp() E[X]=?Var[X]=? Example 7

  24. Summary of Important Moments of Random Variables

  25. Chapter 5 Expectations Expectation of Functions of Multiple Random Variables

  26. The Expectation of Y = g(X1, …, Xn)

  27. Y X Example 8 p(x, y)

  28. Example 9

  29. Chapter 5 Expectations Important Properties of Expectation

  30. 常數之期望值為常數 Linearity E1. E2. X1, X2, …, Xn間不須具備任何條件,上項特性均成立。

  31. Example 10 令X與Y為兩連續型隨機變數,證明E[X+Y] = E[X]+E[Y].

  32. A Question 令X與Y為兩連續型隨機變數,證明E[X+Y] = E[X]+E[Y]. ?

  33. Independence E3. If random variables X1, . . ., Xn are independent, then

  34. Example 11 令X與Y為兩獨立之連續型隨機變數,證明E[XY] = E[X]E[Y].

  35. XY A Question 令X與Y為兩獨立之連續型隨機變數,證明E[XY] = E[X]E[Y]. ?

  36. XY  Example 12

  37. A Question ?

  38. Define The Variance of Sum

  39. The Variance of Sum

  40. The Covariance 差積之期望值

  41. The Covariance

  42. Example 13

  43. XY A Question ?

  44. Properties Related to Covariance E4. E5.

  45. Properties Related to Covariance E4. E5. Fact:

  46. Properties Related to Covariance E4. E5. E6. E7.

  47. Example 14

  48. Example 14

  49. More Properties on Covariance E8.

  50. More Properties on Covariance E8. E9.