Probability theory

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

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

3. 1、Multi-dimensional random variable & distribution function 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

4. definition 3.3 Suppose randm variable series ζ1 (ω), ζ2 (ω),…, ζn (ω) are definited in the same probability space (Ω, F, P ) ，Then ζ (ω)= (ζ1 (ω), ζ2 (ω), …, ζn (ω)) is called a n-dimensional random vector or a n-dimensional random variable.

5. The following function is called the distribution of n-dimensional random variable ζ(ω)= (ζ1(ω),ζ2(ω), …, ζn(ω))

6. 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.

7. 1.1、 Two-dimensional random variable & distribution function DefinitionThe basic space of random experiment E isΩ, ξandηare random variables definited onΩ,then (ξ，η) is called a two-dimensional variable

8. 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

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

10. 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 ．

11. 1.1.1 The distribution function properties of two-dimensional random variable 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 2°F(x，y) is a nondecreasing function for variavles x and y．

12. 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

13. 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

14. Feature 1 Feature 2 Feature3f(x,y) meets the following expression at continuous points

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

16. 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?

17. （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? Solution （1）We have and

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

19. （3）Grapy the regional G={(x,y)|x＞y },and we have

20. 1.3、Several common two-dimensional continuous random variable (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 then (ξ,η) is called to subject to uniform distribution

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

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

23. （2）Marking the following ragionals

24. we have

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

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

27. Have a break !

28. § 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 ξis as follows:

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

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

31. 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.

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

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

34. 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 (2)、continuous random variable

35. Example 10 Suppose (ξ ，η) subject to uniform distribution on region D surrounded by the curve y = x2 and y = x .What are the marginal density functions of random variablesξ 、η. Solution the area of region D is 1 D so the jointed density function of (ξ， η) is

36. 1 D When 0＜x＜1， When x≤0 or x ≥1，

37. Therefore Similarly

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

39. 2．If (ξ，η)~ Find the marginal distributions of ξ、 η.

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

41. 3, The mutual independence of random variables 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.

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

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

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

45. Solution For any x，y, 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. Then

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

47. 1°sufficient condition：It is known that ， then Example 13Suppose (ξ, η)～ ， Proof the necessary and sufficient condition of that ξand η are mutual independent is Proof：

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

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

50. Have a break !