1. Probability Distributions ( 확률분포 ) Chapter 5

2. 2 모든 가능한 ( 확률 ) 변수의 값에 대해 확률을 할당하는 체계 X 가 1, 2, …, 6 의 값을 가진다면 이 6 개 변수 값에 확률을 할당하는 함수 Definition

3. 3 1.Random Variables 2. Binomial Probability Distribution 3. Mean, Var., SD 4. Poisson Distribution Contents

4. 4 1. Random Variables

5. 5 Overview This chapter will deal with the construction of probability distributions by combining the methods of descriptive statistics presented in Chapter 3 and those of probability presented in Chapter 4. Probability Distributions will describe what will probably happen instead of what actually did happen.

6. 6 Definitions  A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure.  A probability distribution is a graph, table, or formula that gives the probability for each value of the random variable.

7. 7  수요수준, 광고효과, 증권수익률 등은 수없이 많은 값을 취할 수 있다. 이러한 값들은 항상 동일한 값을 갖지 않고, 그 값이 수시로 변한다. 이렇게 수시로 변하는 값을 변수 (Variable) 이라 한다.  통계학에서는 변수를 Random Variable ( 확률변수 ) 라 한다. Random Variable ( 확률변수 )

8. 8  Random Variable ( 확률변수 ) 라 부르는 이유는 이 변수가 어느 특정한 값을 취하게 되는 것이 필연적이 아닌 우연적이고 기회적이기 때문이다.  의사결정에 앞서 어떤 행동의 결과, 발생할 수 있는 모든 경우를 고려할 때 각각의 경우는 특정한 순서나 형태로 발생한다기 보다는 확률적으로 발생한다고 볼 수 있으며 따라서 실제로 실험을 해보기 전에는 그 결과를 명확히 알 수 없으므로 이들 결과를 나타내는 변수를 확률변수라 한다. Random Variable ( 확률변수 )

9. 9 Discrete & Continuous Random Variables Discrete random variable either a finite number of values or countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they result from a counting process Continuous random variable infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions

10. 10 The probability histogram is very similar to a frequency histogram, but the vertical scale shows probabilities.

11. 11 Requirements for Probability Distribution P ( x ) = 1 where x assumes all possible values  0  P ( x )  1 for every individual value of x

12. 12 Example p.204 xP(x) 00.2 10.5 20.4 30.3 Table 5-2: Probabilities for a Random Variable Q/ Does Table 5-2 describe a probability distribution?

13. 13 Mean, Variance and Standard Deviation of a Probability Distribution Mean Variance Standard Deviation

14. 14 Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = μ + 2σ Minimum usual value = μ – 2σ

15. 15 Find and. Example xP(x)P(x) 00.5020∙0.502 10.3651∙0.365 20.0982∙0.098 30.0113∙0.011 40.0014∙0.001

16. 16 Identifying Unusual Results with the Range Rule of Thumb. Maximum usual value = μ + 2σ = 3.8+2(1.0)=5.8 Minimum usual value = μ – 2σ = 3.8 - 2(1.0)=1.8 Example 6 (p.211)

17. 17 Identifying Unusual Results Probabilities Rare Event Rule for Inferential Statistics If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct.

18. 18 Identifying Unusual Results Probabilities Using Probabilities to Determine When Results Are Unusual Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) ≤ 0.05. Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer) ≤ 0.05.

19. 19 Expected Value We can think of that mean as the expected value in the sense that it is the average value that we would expect to get if the trials could continue indefinitely. The mean of ______________________ = ______________

20. 20 Expected Value The expected value of a discrete random variable is denoted by E, and it represents the average value of the outcomes. It is obtained by finding the value of [x P(x)].

21. 21 Example 8: Roulette (p.212) Q: If you bet $5 on the number 7 in roulette, what is your expected value of gain or loss? T. 5-4xP(x)P(x) Lose-$537/38-$4.87 Win$(180-5)1/38$4.61 Total-$0.26

22. 22 2. Binomial Probability Distributions ( 이항분포 )

23. 23 Definitions A binomial probability distribution results from a procedure that meets all the following requirements: 1.The procedure has a fixed number of trials. 2.The trials must be independent. 3.Each trial must have all outcomes classified into two categories. 4.The probabilities must remain constant for each trial.

24. 24 Definitions 이항분포란 두 가지 결과를 갖는 실험을 반복시행할 때, 각 시행이 서로 독립적이어서 각 결과가 발생할 확률이 일정할 경우에 성립되는 확률분포이다.

25. 25 Notation for Binomial Probability Distributions S and F (success and failure) denote two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so P(S) = p(p = probability of success) P(F) = 1 – p = q(q = probability of failure) P(x) = p : the probability of getting exactly x successes among n trials.

26. 26 Suppose you are given 4 multiple choice questions, each with 5 possible answers (a, b, c, d, e). Let’s assume that an unprepared student makes random guesses and we want to find the probability of exactly 3 correct responses to the 4 questions. (a) Does this procedure result in a binomial distribution? (b) If this procedure does, identify the values of n, x, p, and q. Example

27. 27 (a)Does this procedure result in a binomial distribution? i. the number of trials _______________. ii. the 4 trials ___________________ iii. each of the 4 trials has ____________: ______________________ iv. for each response, there ______________, which has the probability of a correct response being ________. Example p.198

28. 28 (b) If this procedure does, identify the values of n, x, p, and q. i. with 4 test questions: ___________ ii. we want the probability of exactly 3 correct response: ____________ iii. the probability of success (correct response) for one question is 1/5 [P(S)]: ______________ iv. the probability of failure (wrong response) is 1- P(S): __________ Example p.198

29. 29 Methods for Finding Probabilities We will now present three methods for finding the probabilities corresponding to the random variable x in a binomial distribution. 1.Binomial Probability Formula 2.Binomial Probability Table (Table 1, App. A) 3.Computer Software

30. 30 Method 1: Using the Binomial Probability Formula P(x) = p x q n-x ( n – x ) ! x ! n !n ! for x = 0, 1, 2,..., n where n = number of trials x = number of successes (yes, or head etc) among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p)

31. 31 P(x) = p x q n-x n !n ! ( n – x )! x ! Number of outcomes with exactly x successes among n trials Probability of x successes among n trials for any one particular order Binomial Probability Formula

32. 32 Q: Find the probability of getting exactly 3 correct answers out of 4 multiple-choice questions.  P(3) Example

33. 33 Method 2: Using Table A-1 in Appendix A Part of Table A-1 is shown below. With n = 4 and p = 0.2 in the binomial distribution, the probabilities of 0, 1, 2, 3, and 4 successes are 0.410, 0.410, 0.154, 0.026, and 0.002 respectively.

34. 34 3. Mean, Variance, and Standard Deviation for the Binomial Distribution

35. 35 Probability Distribution Mean Var SD Binomial Distribution

36. 36 This scenario is a binomial distribution where: n = 14 p = 0.5 q = 0.5 Using the binomial distribution formulas: Find the mean and standard deviation for the number of girls in groups of 14 births. : x represents the number of girls in 14 births Example

37. 37 Interpretation of Results Maximum usual values = µ + 2  Minimum usual values = µ – 2  It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits:

38. 38 For this binomial distribution, Determine whether 68 girls among 100 babies could easily occur by chance. Example p.208

39. 39 4. The Poisson Distribution

40. 40  단위시간이나 면적당 특정사건의 발생확률 1. 단위시간당 몰려오는 고객의 수와 관련 (e.g.) 은행창구, 고속도로 통행료 징수소, 슈퍼마켓 계산대, 공항에 도착하는 비행기수 2. 단위면적에 대한 확률변수 (e.g.) 피의 부유물 중 흰피톨의 수, 페인트칠을 한 표면 중 잘 안 칠해진 부분의 비율 Poisson Distribution

41. 41 Poisson Distribution The Poisson distribution applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit.

42. 42 Poisson Distribution x: 단위시간당 사건 : 단위시간당 평균발생률

43. 43

44. 44 Poisson Distribution Requirements  The random variable x is the number of occurrences of an event over some interval.  The occurrences must be random.  The occurrences must be independent of each other.  The occurrences must be uniformly distributed over the interval being used.

45. 45 Mean Var SD Poisson Distribution

46. 46 Example p.235: Earthquakes For a recent period of 100 years, there were 93 major earthquakes in the world. Assume that the Poisson distribution is a suitable model. (a)Find the mean number of major earthquakes per year. The Poisson distribution applies because we are dealing with the occurrences of an event over some interval.

47. 47 Example p.231 (b) If P(x) is the probability of x earthquakes in a randomly selected year, find P(0), P(1), …, P(7). => The prob. of exactly 2 earthquakes in a year is 0.171.

48. 48 Example p.231 (c) The actual results are as follows: 0 major earthquakes: 47 yrs P(0)=0.395: 100*0.395=39.5 yrs 1 major earthquakes: 31 yrs P(1)=0.367: 100*0.367=36.7 yrs 7 major earthquakes: 1 yrs P(7)=0.0000471: 100*0.0000471=0.00471 yrs=1yrs

49. 49  시내 어떤 은행에 매일 10 시와 11 시 사이에 고객이 평균 60 명씩 몰려든다. 10 시와 11 시 사이에 1 분당 2 명이 도착할 확률은 ? P ( x =2) Class Talk

50. 50 Determine if the following is a probability distribution. A.Yes B.No

51. 51 Find the mean of the given probability distribution. A.1.50 B.1.73 C.1.60 D.2.00

52. 52 A contractor is considering a sale that promises a profit of $38,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) 0f $16,000 with a probability of 0.3. What is the expected profit? A. $26,600 B. $22,000 C. $37,800 D. $21,800

53. 53 Use the binomial probability formula to find the probability of x successes in n trials given the probability p of success on a single trial. n = 12, x = 5, p = 0.25 A. 0.103 B. 0.082 C. 0.091 D. 0.027

54. 54 The number of calls received by a car towing service averages 16.8 per day (per 24 hour period). After finding the mean number of calls per hour, use a Poisson Distribution to find the probability that in a randomly selected hour, the number of calls is 2. A. 0.08516 B. 0.13383 C. 0.12166 D. 0.15208