Chapter 3-2 Discrete Random Variables

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