Probability Theory

1. Probability Theory School of Mathematical Science and Computing Technology in CSU Course groups of Probability and Statistics

2. §2.4Figure characteristics of random variables Distribution list is able to describe the statistical characteristics of random variables completely，However, in some practical problems ，only need to know some characteristics of random variables and thus do not need to derive a result of its distribution function.For example： Assessment of the viability of an enterprise, only need to knowthe level of per capita profit of the enterprises； Study the merits of rice varieties, we are concerned aboutthe average rice grainsandthe average weight ofeach piece costs ；

3. Test the quality of cotton, they should not only pay attention tothe average lengthof fiber, but also pay attention tothe deviate degree between the length of fiber and the average length, the longer the average length and the smaller the deviate degree, the better the quality. Study the level of one shooter, we not only depend on hisaverage ring numberwhether high or not, but also depend on hisscope of impactswhether small or not, that is, whether the fluctuations in data small.

4. From the above example we can see，some values relating to random variables. Although we can not completely describe the random variable，but we can clearly describe the important feature of random variables in some respects. Characteristics of these figures have great significance both in theory and practice. One aspect of probability characteristics of random variable are available to describe by figures.

6. Professor consider :the questions are appropriate ,because from the overall look,80 points is representative, the number between people who get more than 80 points and who get less than 80 points are equal. Whose discourse justified? • The average values of random variables——Mathematical expectation • The situation of random variable values are deviate from the mean value on average—— Square The content of this section

7. Section I Mathematical expectation 1.The definition of mathematical expectation Definition: Suppose discrete distribution of random variable X as If the infinite series absolute convergence ,then called which the sum is random variable X as mathematical expectation ,recorded as

8. Example 1 Answer

9. Example 2 Answer

10. B(n,p) np P()  Common mathematical expectation of random variable Distribution Probability distribution Expectation Parameters for the 0-1 distribution of p p

11. 2. The nature of mathematical expectation

12. Prove ：Only prove Nature (4) at n=2

15. Example 7 A Civil Aviation bus contains 20 visitors leave the airport，the visitors can get off at 10 stations，if one station has no passengers to get off the bus will not stop ，take X as the number of stops，Calculate EX（Suppose each passenger get off at various stations have the same possibility, and suppose whether the passengers get off or not are independent of each other ） Answer Import the random variables

16. Then there is

17. That is So （times）

18. Example 8 According to regulation ，one station everyday 8：00 ~ 9:00,9:00 ~ 10:00 both happen to have a bus reach the station，but the reach time is random ，and the arrive time are independent of each other , the law is

19. Answer

22. 3. Themathematical expectation of random variable function

24. Y 0 1 2 3 X 1 0 3/8 3/8 0 3 1/8 0 0 1/8 X 1 3 P 3/4 1/4 Y 0 1 2 3 P 1/8 3/8 3/8 1/8 Example 9 Known the joint distribution of（X,Y）is Answer

26. 3、Simple application of mathematical expectation Blood program selection • For a particular disease survey , n individuals need a blood test , blood tests can be two ways: • Tests separately for each person's blood , need to test n times totally；

27. K individuals will be mixed with the blood tests ，if the test results become negative , then the k individual blood tests only once ；If the results become positive, then the k individuals will have blood test one by one to identify sick persons, then k individual blood tests to be k + 1 times. Suppose the probability of tested positive is p in someone area，and each is a positive person are independent of each other. Try to select a method which can reduce the number of tests. .

28. Xi 1 k + 1 P Answer For the simple calculation ，Based n are multiples of k ，suppose divided into a total of n / k group, the number of tests for group i required to be Xi

29. If Then EX < n Such as，

30. Section II Square Guide example Test the quality of two groups of light bulbs, which were randomly selected 5, the measured lifetime (unit: hours) as follows: A: 2000 1500 1000 500 1000 B: 1500 1500 1000 1000 1000 Let us compare the quality of these two groups of light bulbs

31. Mathematical expectation Square After calculated :Average life are :A:1200 B:1200 After Observated :A has large departure in useful life,B has small departure in useful life,so,B has better quality 1. The definition of square

32. (X - EX)2 —— Random variable X the value of deviation from the average of the situation are a function of X is also a random variable E(X - EX)2 —— Random variable X the value of the average deviation from the average deviation from the degree - a number Note: Variance reflects the random variable relative degree of its deviation from the mean.

33. X is discrete random variables, probability distribution is: If X is continuous random variables, probability density is f (x) Commonly used formula for calculating the variance:

34. 2. The nature of the square

35. 3. Square calculation Example1 Suppose X ~ P (), Calculate DX. Answer

37. Answer TwoImport the random variables Example 2 Suppose X ~ B( n , p)，calculate DX Answer One followthe example above calculateDX

38. are independentof eachother， so,

39. Distribution Probability distribution Square Parameters for the 0-1 distribution of p p(1-p) B(n,p) np(1-p) P()  Common random variable of variance

40. Example 8 Suppose X express the required fire number of shooting independent until hits the target n times. Known for each target shooting in a probability of p，calculate EX , DX Answer X i express the required fire number of hit the target i - 1 times to hit the target i times ，i = 1,2,…, n

41. are independent of each other ,moreover

43. Therefore, Only know the expectations of random variables and the variance can not determine their distribution, such as:

44. -1 0 1 P 0.1 0.8 0.1 -2 0 2 P 0.025 0.95 0.025 They have the same expectation and variance, but the distribution is different . and