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IS 310 Business Statistics CSU Long Beach

IS 310 Business Statistics CSU Long Beach. .40. .30. .20. .10. 0 1 2 3 4. Chapter 5 Discrete Probability Distributions. Random Variables. Discrete Probability Distributions. Expected Value and Variance. Binomial Probability Distribution.

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IS 310 Business Statistics CSU Long Beach

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  1. IS 310 Business Statistics CSU Long Beach

  2. .40 .30 .20 .10 0 1 2 3 4 Chapter 5 Discrete Probability Distributions • Random Variables • Discrete Probability Distributions • Expected Value and Variance • Binomial Probability Distribution • Poisson Probability Distribution

  3. Random Variables A random variable is a variable that can take on values at random. Consider the following experiments: • Asking 10 students if they watched a TV show last night (the number of students who watched the show is a random variable) • Inspecting 20 items of a product to check quality of the items (the number of defective items is a random variable) • Tossing a coin five times (the number of heads occurring is a random variable) • Taking an exam with 100 questions (the number of correct answers is a random variable)

  4. Random Variables (Contd) A random variable can be either Discrete or Continuous Discrete random variables take on certain specific values. Examples are the following: number of defective items in an inspection (0, 1, 2, 3,….); number of correct answers in an exam (0, 1, 2, 3, …); number of heads obtained in tossing a coin five times (0, 1, 2, 3, 4, 5) o---------o---------o---------o---------o The only values the discrete random variable can take on are indicated by circles

  5. Random Variables Contd Continuous Random Variables A continuous random variable can take on any values on a scale. Examples are distance traveled, time taken to go from one place to another, heights of individuals, weights of individuals, temperature of cities, etc. o------------------------------------------o A continuous random variable can take on any value on the above scale

  6. Random Variables Type Question Random Variable x Family size x = Number of dependents reported on tax return Discrete Continuous x = Distance in miles from home to the store site Distance from home to store Own dog or cat Discrete x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)

  7. Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. We can describe a discrete probability distribution with a table, graph, or equation.

  8. Discrete Probability Distribution • Let’s consider the illustration in Section 5.2 (10-Page 190; 11-Page 197) • DiCarlo Motors in Saratoga, New York sold the following number of cars over the past 300 days: • 0 cars on 54 days; 1 car on 117 days; 2 cars on 72 days; 3 cars on 42 days; 4 cars on 12 days; and 5 cars on 3 days. • The probability distribution is shown in Table 5.3 (10-Page 191; 11_Page 198).

  9. Discrete Probability Distribution • Table 5.3 • Number of cars soldProbability • 0 54/300 = 0.18 • 1 117/300 = 0.39 2 72/300 = 0.24 3 42/300 = 0.14 4 12/300 = 0.04 5 3/300 = 0.01

  10. Sample Problem • Problem # 8 (10-Page 193; 11-Page 200) • Number of operating rooms used over a 20-day period. • Number of RoomsFrequency Probability 1 3 3/20 = 0.15 2 5 5/20 = 0.25 3 8 8/20 = 0.40 4 4 4/20 = 0.20

  11. Expected Value and Variance • The expected value of a random variable is obtained by multiplying each value of the random variable by its probability and adding the resulting products. • Let’s refer to the problem of car sales of DiCarlo Motors. Look at Table 5.5 (10-Page 196) or Table 5.4 (11-Page 203). • No. of Cars Sold (x) Probability [f(x)] x. f(x) • 0 0.18 0 • 1 0.39 0.39 • 2 0.24 0.48 • 3 0.14 0.42 • 4 0.04 0.16 5 0.01 0.05 Expected Value of x = E(x) = 1.50

  12. Expected Value and Variance • What does Expected Value mean? • Expected Value is the average value of the random variable over a long period of time. • Referring to DiCarlo Motors, the Expected Value of 1.5 means that DiCarlo can expect to sell, on the average, 1.5 cars per day over a long period of time.

  13. Expected Value and Variance • The variance of a random variable is obtained by using formula 5.5 (10-Page 196; 11-Page 203). Calculations are shown in Table 5.6 (10-Page 197) or Table 5.5 (11-Page 204). • The variance is calculated as 1.25 so the standard deviation is √ 1.25 = 1.118.

  14. Binomial Probability Distribution • Two discrete probability distributions that we will study are: • Binomial Probability Distribution • Poisson Probability Distribution

  15. Binomial Distribution • Four Properties of a Binomial Experiment • 1. The experiment consists of a sequence of n • identical trials. • 2. Two outcomes, success and failure, are possible • on each trial. 3. The probability of a success, denoted by p, does not change from trial to trial. stationarity assumption 4. The trials are independent.

  16. Binomial Distribution Our interest is in the number of successes occurring in the n trials. We let x denote the number of successes occurring in the n trials.

  17. Binomial Distribution • Binomial Probability Function where: f(x) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial

  18. Binomial Distribution • Binomial Probability Function Probability of a particular sequence of trial outcomes with x successes in n trials Number of experimental outcomes providing exactly x successes in n trials

  19. Example of Binomial Distribution • Martin Clothing Store (10-Page 202; 11-Page 209)) • Given: The probability of a customer making a purchase is 0.3. Three customers walk into the store. • What is the probability that two of the three customers will make a purchase? • This is an example of binomial distribution for the following reasons: • 1. There are only two outcomes: making a purchase (success) or not making a purchase (failure). • 2. The probability of success is 0.3 . There are three trials (three customers) and we are trying to determine the probability of two successes.

  20. Example of Binomial Distribution • Martin Clothing Store Problem • Let’s look at Figure 5.3 (10-Page 203; 11-Page 210). • Formula 5.8 (10-Page 205; 11-Page 212) can be used to calculate the probability of two customers making a purchase. • n x n-x • P(x=2) = ( ) p (1 – p) = 0.189 • x

  21. Sample Problem on Binomial Distribution • Martin Clothing Store Problem • Rather than using formula 5.8, we could use Table 5 of Appendix B (10- Pages 930-937; 11-Pages 989-997) to obtain directly the value of any probability without any calculations. We need to know the values of p, x, and n to use Table 5. For x=2, n=3, and p=0.3, the value of P(x=2) = 0.189 from (10-Page 932; 11-Page 992).

  22. Sample Problems Problem # 29 (10-Page 209; 11-Page 216) Given: p = 0.30 x = 3 (number of workers who take public transportation) n = 10 (total number of workers in the sample) a. f(3) = 0.2668 (From Table 5 in Appendix B) b. f(3 or more) = f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9) + f(10) = 0.2668 + 0.2001 + 0.1029 + 0.0368 + 0.0090 + 0.0014 + 0.0001 + 0.0000 = 0.62

  23. Poisson Distribution A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2, . . . ).

  24. Poisson Distribution Examples of a Poisson distributed random variable: the number of knotholes in 14 linear feet of pine board the number of vehicles arriving at a toll booth in one hour

  25. Poisson Distribution • Two Properties of a Poisson Experiment • The probability of an occurrence is the same • for any two intervals of equal length. • The occurrence or nonoccurrence in any • interval is independent of the occurrence or • nonoccurrence in any other interval.

  26. Poisson Distribution • Poisson Probability Function where: f(x) = probability of x occurrences in an interval  = mean number of occurrences in an interval e = 2.71828

  27. Poisson Distribution • Rather than using formula 5.11, one could use Table 7 of Appendix B (10-Pages 939-944; 11-Pages 999-1004) to calculate any probability. We need to know the values of µ and x to use Table 7.

  28. Sample Problem Problem # 40 (10-Page 213; 11-Page 220) a. Given µ = 48 per hour = 4 per five-minute f(3) = 0.1954 (From Table 7 in Appendix B) • Given µ = 12 per 15-minute f(10) = 0.1048 (From Table 7 in Appendix B) • 4 calls f(0) = 0.0183 • f(0) = 0.0907 with µ = 2.4

  29. End of Chapter 5

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