2. Probability theory The department of math of central south university Probability and Statistics Course group

3. 1 、 Multi-dimensional random variable & distribution function 2 、 Multi-dimensional marginal distribution of random variables 3 、 The mutual independence of random variables §3.3 Multi-dimensional random variable

4. When a random phenomenon is considered, many random variables are often needed to be studied,such as launching a shell, the need to study the impact point by several coordinates; study the market supply model, the needs of the supply of goods, consumer income and market Prices and other factors must be taken into account 1 、 Multi-dimensional random variable & distribution function

5. definition 3.3 Suppose randm variable series ζ 1 (ω), ζ 2 (ω),…, ζ n (ω) are definited in the same probability space (Ω, F, P ) ， Then ζ (ω)= (ζ 1 (ω), ζ 2 (ω), …, ζ n (ω)) is called a n-dimensional random vector or a n-dimensional random variable.

6. is called the distribution of n-dimensional random variable ζ(ω)= (ζ 1 (ω),ζ 2 (ω), …, ζ n (ω)) The following function

7. For a n-dimensonal random vector, each of its components is a one-dimensional random variables, and can be separately studied. In addition, most important is that each pair of components are interrelated. We will pay more attention to the two-dimensional random variables. In fact The results about two- dimensional random variables can be applied to multi-dimensional random variables.

8. Definition The basic space of random experiment E isΩ, ξandηare random variables definited onΩ,then (ξ ， η) is called a two-dimensional variable 1.1 、 Two-dimensional random variable & distribution function

9. a Two-dimensional random variable can be ragarded as a random point (ξ ， η) on x-o-y plane space with value （ x ， y ）。 according whether the number of points （ x ， y) that (ξ ， η) get is finite or not,the two-dimensional random variables are divided into two major discrete and continuous random variables

10. is called the distribution function of two- dimensional random variable (ξ ， η) ， or the joint distribution function of ξandη. Definition Suppose (ξ ， η)is a two-dimensional random variable,for any real numbers x ， y ， binary variable function

11. Please pay attention to these rules: 1° {ξ≤x, η ≤ y } express the product event of {ξ ≤x }and {η ≤ y } ． 2° The function value F(x ， y) is the probability that (ξ ， η) get values on the following region : － ∞ ＜ ξ≤ x, － ∞ ＜ η ≤ y ．

12. The distribution function F(x, y) of two dimensional random variable (ξ,η)has the following properties : 1° 0≤F(x,y),and for any number x,y,F(x,y) satisfies 1.1.1 The distribution function properties of two-dimensional random variable 2°F(x ， y) is a nondecreasing function for variavles x and y ．

13. 3°F(x ， y) is left continuous for x and y 4° The probability that (ξ ， η) satisfy x1 ＜ ξ≤x2 ， y1 ＜ η≤y2 is

14. 1.2 、 Two-dimensional continuous random variable Definition Suppose (ξ ， η)is a two-dimensional random variable with distribution function F(x,y) ， if there exists nonnegative function f(x,y) for any x ， y,and F(x,y)satisfies the following integral equation then (ξ ， η) is called a two-dimensional continuous random variable ， f(x,y) is called the joint probability density function of (ξ ， η) ． f(x,y) has the following features

15. Feature 2 Feature 1 Feature 3 f(x,y) meets the following expression at continuous points

16. Feature 4 Let G be a regional of x-o-y plane the probability that points (X, Y) fell within G is

17. Example 6 It is given the probability density function for a two-dimensional random variable (ξ ， η) （ 1 ） What valute is k? （ 2 ） What expression is the distribution function F(x,y)? （ 3 ） What is the probability that ξis large than η?

18. Solution （ 1 ） We have （ 1 ） What valute is k? （ 2 ） What expression is the distribution function F(x,y)? （ 3 ） What is the probability that ξis large than η? and

19. （ 2 ） When x ＞ 0 ， y ＞ 0 So k =6 ． As to other points (x ， y) ， for f (x,y) =0 ， then F(x,y)=0 ． The distribution function can be given as follows:

20. （ 3 ） Grapy the regional G={(x,y)|x ＞ y },and we have

21. (1) ． Uniform distribution Let D be a bound regional in x-o-yplane with area S ， (ξ ， η) is a two-dimensional continuous random variable with density function 1.3 、 Several common two-dimensional continuous random variable then (ξ,η) is called to subject to uniform distribution

22. （ 1 ） What is the probability density function of (ξ,η) ? Example 7 A two –dimensional random variable (ξ,η) subjects to uniform in region （ 2 ） What is the probability that (ξ,η) gets value in region

23. Solution （ 1 ） Draw the graph of regional D and acounting the area,then the probability density function of (ξ,η) is given as follows

24. （ 2 ） Marking the following ragionals

25. we have

26. Then (ξ ， η) is subjected to a two-dimensional normal distribution with parameters (2) ． Normal distribution Suppose (ξ ， η) is a two-dimensional random variable wih probability density function Here are constants ， and and denoted as (ξ,η) ~

27. Example 8 suppose (ξ ， η) is a two-dimension random variable with density function Solution ． Please calculate

28. Have a break !

29. § 3.3 the distribution of multi-dimensional random variables (continued) 2 、 the marginal distribution of two-dimensional random variables Given the distribution function F(x ， y) of (ξ ， η) ， then the marginal distribution function of random variable ξ is as follows:

30. (1)Discrete random variables Then It is known the joint distribution law of random variable (ξ ， η) in the following the marginal distribution function of random variable η can also be expressed as follows: Let’s racall the marginal distribution of discrete raandom variables.

31. That is,the marginal distribution law of random variable ξ can be expressed as Similarly,the marginal distribution law of random variable ηis as follows

32. Example 9 The joint distribution law of (ξ ， η) is as follows. η ξ 1 2 1 1 / 6 0 2 0 1 / 6 3 1 / 6 0 4 0 1 / 6 5 1 / 6 0 6 0 1 / 6 Please calculate the marginal distribution law of random variable of ξ 、 η,respectively.

33. Solution ξ 1 2 3 4 5 6 P 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 … So we can easily outline the distribution law of ξ in the following tableau

34. The marginal distribution of η is : Y 1 2 P 1 / 2 1 / 2

35. (2) 、 continuous random variable Random variable (ξ ， η) has jointed density function f (x, y) ， then the marginal distribution funtion of ξ can be expressed as the marginal probability density function of ξ is With the same ， the marginal probability density of η is

36. Example 10 Suppose (ξ ， η) subject to uniform distribution on region D surrounded by the curve y = x 2 and y = x.What are the marginal density functions of random variablesξ 、 η. 1 D Solution the area of region D is so the jointed density function of (ξ ， η) is

37. When 0 ＜ x ＜ 1 ， 1 D When x≤0 or x ≥1 ，

38. Therefore Similarly

39. 1 ． Suppose (ξ ， η) is subjected to uniform dis tribution on a region surrounded by linears x=0 ， y=0 ， x+y=1.Find the marginal distribution of r andom variablesξ 、 η ． Exercise

40. 2 ． If (ξ ， η)~ Find the marginal distributions of ξ 、 η.

41. The marginal density functions of ξ 、 η are outlined as follows,respectivily. ， ． That is,

42. Definition (ξ ， η)is a two –dimensional random variable ， if the joint distribution of (ξ ， η) equal the product of marginal distribution of ξ and η, then ξ and η are independent of each other. 3, The mutual independence of random variables

43. If is the jointed distribution of (ξ ， η), F ξ ( x) 、 F η (y) are the marginal distribution function of ξ 、 η,respectivily,then the necessary and sufficient conditions of thatξ and η are mutually independent is

44. Especially,For a two-dimensional discrete random variable (ξ ， η) ， then the necessary and sufficient conditions of thatξ and η are mutually independent is.

45. Moreover,for a two-dimensional continuous random variable (ξ ， η) ， then the necessary and sufficient conditions of thatξ and η are mutually independent is

46. Example 11 A two-dimensional random variable (ξ ， η) has probability density function as follows Judge whether ξ,ηare mutual independent or not Based on this point,we know that the ξ,ηare mutual independent. Solution For any x ， y, Then

47. Solution. Example 12 Suppose ξ and ηare mutual independent, Calculate the probability of

48. Proof ： 1°sufficient condition ： It is known that ， then Example 13 Suppose (ξ, η) ～ ， Proof the necessary and sufficient condition of that ξand η are mutual independent is

49. and ， so ， Which means ξand ηare independent ．

50. Therefore ． Especially,let, We can get the following equation, 2°Necessary condition ： It is known that ξand ηare independent,then for any number x ， y,the following equation is established

51. Have a break !