# CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS

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1 CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS TRUE/FALSE 235. The Poisson probability distribution is a continuous probability distribution. F 236. In a Poisson distribution, the mean and variance are equal. T 237. The Poisson random variable is a discrete random variable with infinitely many possible values. T 238. The mean of a Poisson distribution, where is the average number of successes occurring in a specified interval, is. T 239. The number of accidents that occur at a busy intersection in one month is an example of a Poisson random variable. T 240. The number of customers arriving at a department store in a 5-minute period has a Poisson distribution. T 241. The number of customers making a purchase out of 30 randomly selected customers has a Poisson distribution. F 242. The largest value that a Poisson random variable X can have is n. F 243. The Poisson distribution is applied to events for which the probability of occurrence over a given span of time, space, or distance is very small.

2 T 244. In a Poisson distribution, the variance and standard deviation are equal. F 245. In a Poisson distribution, the mean and standard deviation are equal. F MULTIPLE CHOICE 246. Which of the following cannot have a Poisson distribution? a. The length of a movie. b. The number of telephone calls received by a switchboard in a specified time period. c. The number of customers arriving at a gas station in Christmas day. d. The number of bacteria found in a cubic yard of soil. A 247. The Sutton police department must write, on average, 6 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day. Interpret the value of the mean. a. The mean has no interpretation. b. The expected number of tickets written would be 6.5 per day. c. Half of the days have less than 6.5 tickets written and half of the days have more than 6.5 tickets written. d. The number of tickets that is written most often is 6.5 tickets per day. B 248. The Poisson random variable is a: a. discrete random variable with infinitely many possible values. b. discrete random variable with finite number of possible values. c. continuous random variable with infinitely many possible values. d. continuous random variable with finite number of possible values. A 249. Given a Poisson random variable X, where the average number of successes occurring in a specified interval is 1.8, then P(X = 0) is: a. 1.8 b c d C

3 250. In a Poisson distribution, the: a. mean equals the standard deviation. b. median equals the standard deviation. c. mean equals the variance. d. None of these choices. C 251. On the average, 1.6 customers per minute arrive at any one of the checkout counters of Sunshine food market. What type of probability distribution can be used to find out the probability that there will be no customers arriving at a checkout counter in 10 minutes? a. Poisson distribution b. Normal distribution c. Binomial distribution d. None of these choices. A 252. A community college has 150 word processors. The probability that any one of them will require repair on a given day is To find the probability that exactly 25 of the word processors will require repair, one will use what type of probability distribution? a. Normal distribution b. Poisson distribution c. Binomial distribution d. None of these choices. C COMPLETION 253. In a Poisson experiment, the number of successes that occur in any interval of time is of the number of success that occur in any other interval. independent 254. In a(n) experiment, the probability of a success in an interval is the same for all equal-sized intervals. Poisson 255. In a Poisson experiment, the probability of a success in an interval is to the size of the interval. proportional

4 256. In Poisson experiment, the probability of more than one success in an interval approaches as the interval becomes smaller. zero A Poisson random variable is the number of successes that occur in a period of or an interval of in a Poisson experiment. time; space 258. The of a Poisson distribution is the rate at which successes occur for a given period of time or interval of space. mean expected value 259. In the Poisson distribution, the mean is equal to the. variance 260. In the Poisson distribution, the is equal to the variance. mean 261. The possible values of a Poisson random variable start at. zero 0

5 262. A Poisson random variable is a(n) random variable. discrete SHORT ANSWER 263. Compute the following Poisson probabilities (to 4 decimal places) using the Poisson formula: a. P(X = 3), if = 2.5 b. P(X 1), if = 2.0 c. P(X 2), if = 3.0 a b c Let X be a Poisson random variable with = 6. Use the table of Poisson probabilities to calculate: a. P(X 8) b. P(X = 8) c. P(X 5) d. P(6 X 10) a b c d Let X be a Poisson random variable with = 8. Use the table of Poisson probabilities to calculate: a. P(X 6) b. P(X = 4) c. P(X 3) d. P(9 X 14) a

6 b c d NARRBEGIN: 911 Phone Calls 911 Phone Calls 911 phone calls arrive at the rate of 30 per hour at the local call center. NARREND 266. {911 Phone Calls Narrative} Find the probability of receiving two calls in a five-minute interval of time. = 5(30/60) = 2.5; P(X = 2) = {911 Phone Calls Narrative} Find the probability of receiving exactly eight calls in 15 minutes. = 15(30/60) = 7.5; P(X = 8) = {911 Phone Calls Narrative} If no calls are currently being processed, what is the probability that the desk employee can take four minutes break without being interrupted? = 4(30/60) = 2.0; P(X = 0) = NARRBEGIN: Classified Department Pho Classified Department Phone Calls A classified department receives an average of 10 telephone calls each afternoon between 2 and 4 P.M. The calls occur randomly and independently of one another. NARREND 269. {Classified Department Phone Calls Narrative} Find the probability that the department will receive 13 calls between 2 and 4 P.M. on a particular afternoon. = 10; P(X = 13) = 0.072

7 270. {Classified Department Phone Calls Narrative} Find the probability that the department will receive seven calls between 2 and 3 P.M. on a particular afternoon. = 5; P(X = 7) = {Classified Department Phone Calls Narrative} Find the probability that the department will receive at least five calls between 2 and 4 P.M. on a particular afternoon. = 10; P(X 5) = NARRBEGIN: Post office Post office The number of arrivals at a local post office between 3:00 and 5:00 P.M. has a Poisson distribution with a mean of 12. NARREND 272. {Post Office Narrative} Find the probability that the number of arrivals between 3:00 and 5:00 P.M. is at least 10. =12; P(X 10) = {Post Office Narrative} Find the probability that the number of arrivals between 3:30 and 4:00 P.M. is at least 10. = 3; P(X 10) = {{Post Office Narrative} Find the probability that the number of arrivals between 4:00 and 5:00 P.M. is exactly two. = 6; P(X = 2) = 0.045

8 275. Suppose that the number of buses arriving at a Depot per minute is a Poisson process. If the average number of buses arriving per minute is 3, what is the probability that exactly 6 buses arrive in the next minute? NARRBEGIN: Unsafe Levels of Radioact Unsafe Levels of Radioactivity The number of incidents at a nuclear power plant has a Poisson distribution with a mean of 6 incidents per year. NARREND 276. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be exactly 3 incidents in a year {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be at least 3 incidents in a year {Unsafe Levels of Radioactiviy Narrative} Find the probability that there will be at least 1 incident in a year {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be no more than 1 incident in a year

9 280. {Unsafe Levels of Radioactivity Narrative} Find the variance of the number of incidents in one year {Unsafe Levels of Radioactivity Narrative} Find the standard deviation of the number of incidents is in one year. 2.45

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