Probability Theory

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

2. Chapter 2 Discrete type random variable §2.1 One-dimensional random variable and distribution §2.2 Multiple random variables and distribution §2.3 The distribution of the function of random variable §2.4 Numerical characteristic of random variable §2.5 Conditional distribution and conditional mathematical expectation

3. In order to better reveal the regularity of random phenomenon, and to use mathematical tools to describe its rule，we introduce stochastic variable to describe the different results of randomized trials. E.g. The receiving phone numbers of telephone switchboard over a long period can be described by variable X E.g. Two results of throwing a coin can also be described by a variable

4. §2.1 One-dimensional random variable and distribution 1、The concept of random variables E.g. (1) Randomly throw a dice，ω said all the sample points, ω: One point Two point Three point Four point Five point Six point X(ω): 1 2 3 4 5 6

5. (2) Someone continuously shoot on the same target, until really shot and then stop. ω is the number of shots，so ω One shooting Two shooting.... n shooting...... X(ω) 1 2 .... n ...... (3) Every 10 minutes a bus is driven from one station, passengers at any time get to the station, and ωis the passenger waiting time, ωWaiting time X(ω) [0, 10]

6. 1、The concept of discrete random variables Define If the possible value of the random variable X is limited or infinite list，say X for discrete random variables, generally useX, Y , Z ,or lowercase letters Greece , ,  to show. Description of probability characteristic of discrete random variables commonly use its probability distribution or distribution law,

7. nonnegative standard The characteristic of probability distribution

8. Random variablesis a mapping， The characteristic of this mapping as follows： Field of definitions: Randomicity: Random variable X possible value is more than one，before the test can only predict its possible values but cannot predict which value is

9. Probability characteristic: X to take a certain probability to take someone value or some values After introducing random variables, using random variable equation or inequality to express random events

10. For example，If use X to depict the receiving phone numbers of telephone switchboard at 9:00-10:00, or —— is “Some days at 9:00 ~ 10:00 the receiving telephone numbers is more than 100 times”

11. Such as，use random variables to describe the possible results of throwing a coin,so

12. — Positive upward also describe the randomized trial results

13. Such as，To study the development on children's growing often require multiple indexes，the height、weight、head circumference etc.  = {the development on children's growing } X (  ) —height Y (  ) —weight Z (  ) —head circumference The random variables may have a certain relationship，or may have not a certain relationship——be Independent to each other

14. note： The probability distribution of the discrete random variables can be gotten as following steps: (1) Determine all possible values of random variables; (2) Try to（Using the classical probability）calculate the probability to take each value. (3) List the probability distribution of random variables（or write probability function）.

15. Example 1 From 1 ~ 10 this 10 numbers randomly take out of five figures，and make X：remove the maximum number of five figures. Try to get the distribution law for X． Getting the distribution rate must explain k value range! Solution: The possible value for X is 5,6,7,8,9,10. And =——

16. x 5 6 7 8 9 10 P Concretely writing，and get the distribution law for X：

17. Example 2 5 black balls and 3 white balls within a bag, each one taken does't return every time, until the ball for black. Remember X as the numbers of taking white balls, rememberY as the numbers of taking balls，Get the probability distribution for X, Y and the probability for taking at least three times. Solution (1) The possible value for X is 0,1,2,3, P(X=0)=5/8, P(X=1)=(3×5)/(8×7)=15/56 P(X=2)=(3×2×5)/(8 ×7 ×6)=5/56, P(X=3)=1/56, so, The probability distributionof X is

18. X 0 1 2 3 P 5/8 15/56 5/56 1/56 (2) The possible value for Y is1,2,3,4, P(Y=1)=5/8, P(Y=2)=P(X=1)=15/56, similarly P(Y=3)=P(X=2)=5/56, P(Y=4)=P(X=3)=1/56, So the probability distribution of Y is

19. Y 1 2 3 4 P 5/8 15/56 5/56 1/56 (3) P(Y≥3)=P(Y=3)+P(Y=4)=6/56

20. 0 < p <1 X = xk1 0 Pk p 1 - p Third. Common discrete random variables distribution (1) 0 – 1 distribution

21. Application All randomized trials only have two possible results，usually described by 0-1 distribution，such as whether the product is qualified, the population gender statistic, whether the system is normal, and whether the consumption of power is overloaded etc. Note The distribution law can be written as

22. As in “dice” experiment，useto depict｛i appear｝，random variables X is even distribution. (2)Discrete even distribution

24. (3) Binomial distribution Background：In N heavy Bernoulli trial， the times of A interested in each test of n tests—— X is a discrete random variable If P ( A ) = p , so

25. Say X is the binomial distribution and parameter is n, p，remember as 0 – 1 distribution is binomial distribution (n = 1)

26. Example 2 The defective product rate in a large number of product is 0.1, now one is scoopped out. Try to get the the probability as following events： B={Just 2 defective products from removing 15 products} C={At least 2 detective products from removing 15 products} Solution：Because of removing 15 productsin a large number products，so it can be similarly regarded as a 15 heavy Bernoulli test．

27. So，

28. Example 3: A person who don't know English at all attend a English exam. Supposed this exam has 5 multiple-choice, every problem has n choices, only one answer is correct. Try to: the probability of the event that he can answer three questions above and then pass.

29. Solution: Because he completely didn't know，So every answer in every question is the same for him， and whether he correctly answered the question also is independent. So, the process of his answering is the Bernoulli test.

31. or (4) Poisson Distribution If is constant，Say the parameters for X is for PoissonDistribution,as or Application In a certain time intervals： The receiving numbers of phone from telephone switchboard；

32. The numbers of defects on the cloth； The numbers of customers in shopping malls； The numbers of patinents in Municipal hospital emergency； The numbers of bacteria in a container； The numbers of traffic accidents in one area; The numbers of particles from radioactive materials； The numbers of printing mistakes from every page in a book； ……

33. Example 4 Set random variable X obey the Poisson distribution and the the parameters isλ，and Solution： The distribution law for random variable X is Because

34. Example 5 If the distribution law of the random variable X is Try to get the constant c unknown . Soulation: The nature of the distribution law

35. (5)Geometric distribution Set the probability of a machine gun shotting down a aircraft for p,Unlimited firing，so the numbers needed of first shotting down plane , the distribution of X obey geometric distribution ,and the parameters is p，written as ,so

36. Easily prove,if the airplanes didn't shotted down before m times,so,in the condition，In order to make the waitting time, before plane was shotted down, also obey the same geometric distribution，the distribution is irrelevant with m，This is the so-called no memory.

37. With S products,there are N good products，and M defective products( )，randomly take n products and don't put back， write X as the numbers of the good products taken，get the distribution law for X. Now the probability of taking k good products： (6)Super geometry distribution

38. k = 0，1，… ，n so X obey super geometry distribution.as

39. It can be proved that limit distribution of super geometry distribution was binomial distribution,in practical application, when all S，M，N, arelarge，super geometry distribution can be approximately depicted as follow

40. (7) Negative binomial distribution（Pascal distribution) (Self-study) (8) Zipfdistribution(Self-study)

41. Have a break and then go on