2. Free Powerpoint Templates ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS ROHANA BINTI ABDUL HAMID INSTITUTE FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS

3. PROBABILITY DISTRIBUTION CHAPTER 3

4. PROBABILITY DISTRIBUTION 3.1 Introduction 3.2 Binomial distributi on 3.3 Poisson distributi on 3.4 Normal distributio n

5. 3.1 INTRODUCTION A probability distribution is obtained when probability values are assigned to all possible numerical values of a random variable. Probability distribution can be classified either discrete or continuous.

6. BINOMIAL DISTRIBUTION POISSON DISTRIBUTION DISCRETE DISTRIBUTIO NS NORMAL DISTRIBUTION CONTINUOUS DISTRIBUTIO NS

7. 3.2 THE BINOMIAL DISTRIBUTION Definition 3.1 : An experiment in which satisfied the following characteristic is called a binomial experiment: 1. The random experiment consists of n identical trials. 2. Each trial can result in one of two outcomes, which we denote by success, S or failure, F. 3. The trials are independent. 4. The probability of success is constant from trial to trial, we denote the probability of success by p and the probability of failure is equal to (1 - p) = q.

8. Examples of Binomial Distribution 1. A university found that 20% of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course. Find the probability…. 2. In Kuala Lumpur, 30% of workers take public transportation daily. In a sample of 10 workers,…. 3. Suppose 20% of the marbles packed in a box are red in color. Suppose 4 marbles are chosen at random. Find the probability of …

9. 1) Calculate probability using formula Definition 3.2 : A binomial experiment consist of n identical trial with probability of success, p in each trial. The probability of x success in n trials is given by  x = 0, 1, 2,......, n

10.  Table of binomial can be used to find the probabilities using the following rules as the guidelines. 2) Calculate probability using table

11. Definition 3.3 :The Mean and Variance of X If X ~ B(n,p), then where  n is the total number of trials,  p is the probability of success and  q is the probability of failure. MeanVariance Standard deviation

12. E XAMPLE 3.1 Given that X~B(12, 0.4), find

13. SOLUTIONS Try to find the probabilities using table. Do you get the same answer???

15. Powerpoint Templates Page 14 Exercise In Kuala Lumpur, 30% of workers take public transportation daily. In a sample of 10 workers, I.What is the probability that exactly three workers take public transportation daily? II.What is the probability that at least three workers take public transportation daily? III.Calculate the standard deviation of this distribution.

16. Powerpoint Templates Page 15 Solution X=number of workers take public transportation daily.

17. Powerpoint Templates Page 16

18. Powerpoint Templates Page 17 Extra Exercise In a large shipment of automobile tires, 5% have a certain flaw. A sample of four tires was chosen to be installed on a car. Let X be the random variable of tires have flaw. a)What is probability that at least one of the tires has a flaw? b)What is the probability that exactly three of the tires have no flaw?

19. 3.3 The Poisson Distribution Definition 3.3  A random variable X has a Poisson distribution and it is referred to as a Poisson random variable if and only if its probability distribution is given by

20.  λ (Greek lambda) is the long run mean number of events for the specific time or space dimension of interest.  A random variable X having a Poisson distribution can also be written as

21. REMEMBER!!!!!!!!!!!!!!!!!!!! Average Rate Mean Average Rate Mean

22. E XAMPLE 3.2 Given that, find

23. SOLUTIONS Try to find the probabilities using table. Do you get the same answer???

24. E XAMPLE 3.3 Suppose that the number of errors in a piece of software has a Poisson distribution with parameter. Find a) the probability that a piece of software has no errors. b) the probability that there are three or more errors in piece of software. c) the mean and variance in the number of errors.

25. SOLUTIONS

26. Powerpoint Templates Page 25 Exercise 1 Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways I.Find the probability of receiving three calls in a 5-minutes interval time. II.Find the probability of receiving more than two calls in 15 minutes.

27. Powerpoint Templates Page 26 Exercise 2 An average of 15 aircraft accidents occurs each year. Find I.The mean, variance and standard deviation of aircraft accident per month. II.The probability of no accident during a months.

28. IMPORTANT!!!! exactly two= 2

29. More than two/ Exceed two Two or more/ At least two/ Two or more

30. less than two/ Fewer than two At most two/ Two or fewer/ Not more than two

31. BINOMIAL DISTRIBUTION POISSON DISTRIBUTION DISCRETE DISTRIBUTIO NS NORMAL DISTRIBUTION CONTINUOUS DISTRIBUTIO NS

32. 3.4 NORMAL DISTRIBUTION 3.4.3 NORMAL APPROXIMATION OF THE POISSON DISTRIBUTION 3.4.2 NORMAL APPROXIMATION OF THE BINOMIAL DISTRIBUTION 3.4.1 INTRODUCTION

33. Definition 3.4

34. Normal Distribution

35. The Normal Distribution has:  mean = median = mode  symmetry about the center  50% of values less than the mean and 50% greater than the mean

36. Applications of normal distribution  Many naturally occurring random processes tend to have a distribution that is approximately normal. Examples can be found in any field, these include:  heights and weights of adults  length and width of leaves of the same species  actual weights of rice in 5 kg bags sold in supermarkets

37. The Standard Normal Distribution  The normal distribution with parameters and is called a standard normal distribution.  A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by.

38. Standardizing A Normal Distribution

39. Why Standardize... ?  It can help you make decisions about your data.  It also makes life easier because we only need one table (the Standard Normal Distribution Table), rather than doing calculations individually for each value of mean and standard deviation.

40. Example: Professor Willoughby is marking a test.  Here are the students results (out of 60 points): 20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17  Most students didn't even get 30 out of 60, and most will fail.The test must have been really hard, so the Prof decides to Standardize all the scores and only fail people 1 standard deviation below the mean.  The Mean is 23, and the Standard Deviation is 6.6, and these are the Standard Scores:  -0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, - 0.91  Only 2 students will fail (the ones who scored 15 and 14 on the tes t)

42. Standard normal distribution Total area =1

43. E XAMPLE 3.1 Determine the probability or area for the portions of the Normal distribution described. (using the normal table)

44. SOLUTIONS Using table

48. C ALCULATE THE PROBABILITIES USING CALCULATOR - Mode: SD -1 - Shift 3 - P(1) for - Q (2) for - R (3) for

49. E XAMPLE 4 Determine the probability or area for the portions of the Normal distribution described. (using the calculator)

51. The masses of a well known brand of breakfast cereal are normally distributed with mean of 250g and standard deviation of 4g. Find the probability of a packet containing more than 254.4g. E XAMPLE 3.2

52. Let X be the r.v. “masses of cereal in grams” where X~N(250, 16). SOLUTIONS

53. EXERCISE 1 A battery has a lifetimes which are normally distributed with a mean of 62 hours and a standard deviation of 3 hours. What is the probability of battery lasting less than 68 hours?

54. EXERCISE 2 A carton of orange juice has a volume which is normally distributed with a mean of 120ml and a standard deviation of 1.8ml. Find the probability that the volume is more than 118ml.

55. EXERCISE 3 The pulse rate is a measure of the number of heart beats per minute. Suppose that the pulse rates for adults are assumed to be normally distributed with a mean of 78 and a standard deviation of 12. Find the probability that adults will have the pulse rates between 60 and 100.

56. Find the value of Z

57. Example 3.3

58. SOLUTIONS Using table

61. In January 2003, the American worker spent an average of 77 hours logged on to the internet while at work. Assume that the population mean is 77 hours, the times are normally distributed, and the standard deviation is 20 hours. A person is classified as heavy user if he or she is in the upper 20% of usage. How many hours did a worker have to be logged on to be considered a heavy user? Example 3.4

62. SOLUTIONS Let X be the r.v. “hours of worker spent on internet” where X~N(77, 20 2 ).

63. 3.4 NORMAL DISTRIBUTION 3.4.3 NORMAL APPROXIMATION OF THE POISSON DISTRIBUTION 3.4.2 NORMAL APPROXIMATION OF THE BINOMIAL DISTRIBUTION 3.4.1 INTRODUCTION

64. 3.4.2 Normal Approximation of the Binomial Distribution  When the number of observations or trials n in a binomial experiment is relatively large, the normal probability distribution can be used to approximate binomial probabilities. A convenient rule is that such approximation is acceptable when

66. Definiton 3.5

67. Continuous Correction Factor  The continuous correction factor needs to be made when a continuous curve is being used to approximate discrete probability distributions.

68. Example 3.5 In a certain country, 45% of registered voters are male. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males.

69. Solutions Let X be the r.v. “number of male voters” where X~B(300, 0.45).

71. 3.4 NORMAL DISTRIBUTION 3.4.3 NORMAL APPROXIMATION OF THE POISSON DISTRIBUTION 3.4.2 NORMAL APPROXIMATION OF THE BINOMIAL DISTRIBUTION 3.4.1 INTRODUCTION

72. 3.4.3 Normal Approximation of the Poisson Distribution  When the mean of a Poisson distribution is relatively large, the normal probability distribution can be used to approximate Poisson probabilities.  A convenient rule is that such approximation is acceptable when  Definition 3.6

74. Example 3.6 A grocery store has an ATM machine inside. An average of 5 customers per hour comes to use the machine. What is the probability that more than 30 customers come to use the machine between 8.00 am and 5.00 pm?

75. Solutions Let X be the r.v. “number of customers per hour” where X~P 0 (5). Let X be the r.v. “number of customers for 9 hours” where X~P 0 (45).

77. 3.4.4 Poisson Approximation of the Binomial Distribution  When the number of observations or trials n in a binomial experiment is relatively large (n≥30) and probability of success is very small (np<5), the Poisson probability distribution can be used to approximate binomial probabilities.  A convenient rule is that such approximation is acceptable when

78. Example 3.7 Suppose a life insurance company insures the lives of 5000 men aged 42. If actual studies show that the probability that any 42-year-old man will die in a given year to be 0.001, find the exact probability that the company will have to pay more than 10 claims during a given year.

79. Solutions Let X = number of claims/number of any 42-year- old man will die in a given year where X~B(4000,0.001). Using binomial distribution; P(X>10) Not available in binomial tables!!!!

80. use Poisson approximation since Look at Poisson table!

81. EXERCISE 1 According to a survey by Duit magazine, 27% of women expect to support their parents financially. Assume that this percentage holds true for the current population of all women. Suppose that a random sample of 300 women is taken. Find the probability that exactly 79 of the women in this sample expect to support their parents financially.

82. EXERCISE 2 Aonang Beach Resort Hotel has 120 rooms. In the spring months, hotel room occupancy is approximately 75%. I. What is the probability that 100 or more rooms are occupied on a given day. II. What is the probability that 80 or fewer rooms are occupied on a given day?

83. EXERCISE 3 In a university, the average of the students that come to the student health center is 5 students per hour. What is the probability that at least 40 students will come to the student health center from 9.00 am to 6.00 pm?

84. EXERCISE 4 Suppose that at a certain automobile plant the average number of work stoppages per day due to equipment problems during the production process is 12.0. What is the approximate probability of having 15 or fewer work stoppages due to equipment problems on any given day?

85. The probability that a person will develop an infection even after taking a vaccine that was supposed to prevent the infection is 0.003. In a random sample of 200 people in a community who got the vaccine, what is the probability that two or fewer people will be infected? EXERCISE 5