1 / 99

Chapter 3-2 Discrete Random Variables

Chapter 3-2 Discrete Random Variables. 主講人 : 虞台文. Content. Functions of a Single Discrete Random Variable Discrete Random Vectors Independent of Random Variables Multinomial Distributions Sums of Independent Variables  Generating Functions Functions of Multiple Random Variables.

meryle
Télécharger la présentation

Chapter 3-2 Discrete Random Variables

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 3-2Discrete Random Variables 主講人:虞台文

  2. Content • Functions of a Single Discrete Random Variable • Discrete Random Vectors • Independent of Random Variables • Multinomial Distributions • Sums of Independent Variables  Generating Functions • Functions of Multiple Random Variables

  3. Chapter 3-2Discrete Random Variables Functions of a Single Discrete Random Variable

  4. 計程車司機的心聲 這傢伙上車後會要跑幾公里(X)? X為一隨機變數

  5. 隨機變數之函式亦為隨機變數。 Y = g(X) 計程車司機的心聲  這傢伙上車後會要跑幾公里(X)? X為一隨機變數 Y亦為一隨機變數 這傢伙上車後我可以從他口袋掏多少錢(Y)?

  6. Y = g(X) 若pX(x)已知, pY(y)=? 計程車司機的心聲  這傢伙上車後會要跑幾公里(X)? X為一隨機變數 Y亦為一隨機變數 這傢伙上車後我可以從他口袋掏多少錢(Y)?

  7. The Problem Y = g(X)and pX(x)is available.

  8. 福氣啦!!! 這瓶只要五元 這瓶十元 Example 17

  9. 福氣啦!!! 這瓶只要五元 這瓶十元 Example 17

  10. 福氣啦!!! 這瓶只要五元 這瓶十元 Example 17

  11. Example 17

  12. Example 18 n=10, p=0.2.

  13. Example 18 n=10, p=0.2.

  14. Example 18 n=10, p=0.2.

  15. Pay 100$, #bottles (X3) obtained? Example 18 n=10,p=0.2.

  16. Pay 100$, #bottles (X3) obtained? Example 18 n=10,p=0.2. Let Y (X3) denote #lucky bottles obtained.

  17. Chapter 3-2Discrete Random Variables Discrete Random Vectors

  18. Definition  Random Vectors A discrete r-dimensional random vectorX is a function X:   Rr with a finite or countable infinite image of {x1, x2, …}.

  19. Example 19

  20. 1 Example 19

  21. 2 Example 19

  22. Definition  Joint Pmf Let random vector X = (X1, X2, …, Xr). The joint pmf (jpmf) for X is defined as pX(x) = P(X1 = x1, X2 = x2, … , Xr = xr), where x = (x1, x2, … , xr).

  23. Y X Example 20 There are three cards numbered 1, 2 and 3. Randomly draw two cards among them without replacement. Let X, Y represent the number of the 1st and 2nd card, respectively. Find the jpmf of X, Y.

  24. Y X Example 20 There are three cards numbered 1, 2 and 3. Randomly draw two cards among them without replacement. Let X, Y represent the number of the 1st and 2nd card, respectively. Find the jpmf of X, Y.

  25. Properties of Jpmf's • p(x)  0, x Rr; • {x | p(x)  0}is a finite or countably infinite subset ofRr;

  26. Let X = (X1, …, Xi, …, Xr) be an r-dimensional random vectors. The ithmarginal probability mass function defined by Definition Marginal Probability Mass Functions

  27. Y X Example 21 Find pX(x) and pY (y) of Example 20.

  28. Y X Example 21 Find pX(x) and pY (y) of Example 20.

  29. 4 X =# Y =# • pX,Y(x, y) =? • pX(x) =?pY(y) =? • p(X < 3)= ? • p(X + Y < 4)= ? Example 22

  30. 4 pX,Y(x, y) X =# Y =# • pX,Y(x, y) =? • pX(x) =?pY(y) =? • p(X < 3)= ? • p(X + Y < 4)= ? Example 22

  31. 4 pX,Y(x, y) X =# Y =# • pX,Y(x, y) =? • pX(x) =?pY(y) =? • p(X < 3)= ? • p(X + Y < 4)= ? Example 22

  32. 4 pX,Y(x, y) X =# Y =# • pX,Y(x, y) =? • pX(x) =?pY(y) =? • p(X < 3)= ? • p(X + Y < 4)= ? Example 22

  33. 4 pX,Y(x, y) X =# Y =# • pX,Y(x, y) =? • pX(x) =?pY(y) =? • p(X < 3)= ? • p(X + Y < 4)= ? Example 22

  34. 4 pX,Y(x, y) X =# Y =# • pX,Y(x, y) =? • pX(x) =?pY(y) =? • p(X < 3)= ? • p(X + Y < 4)= ? Example 22

  35. Chapter 3-2Discrete Random Variables Independent Random Variables

  36. Let X1, X2, …, Xr be r discrete random variables having densities , respectively. These random variables are said to be mutually independent if their jpdf p(x1, x2, …, xr) satisfies Definition

  37. Example 23 Tossing two dice, let X, Y represent the face values of the 1st and 2nd dice, respectively. 1. pX,Y (x, y) = ?. 2. Are X, Yindependent?

  38. Example 23

  39. Fact  ? ? ?

  40. Fact

  41. Fact

  42. Example 24 Consider Example 23. Find P(X  2, Y 4).

  43. Example 24

  44. Example 24

  45. Example 24 Z1有何意義?

  46. Example 24

  47. Example 24

  48. Example 24

  49. Example 24 p’ p’

  50. Example 24

More Related