CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS

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CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS

TRUE/FALSE

235. The Poisson probability distribution is a continuous probability distribution.

ANS: F PTS: 1 REF: SECTION 7.5

NAT: Analytic; Probability Distributions

236. In a Poisson distribution, the mean and variance are equal.

ANS: T PTS: 1 REF: SECTION 7.5

237. The Poisson random variable is a discrete random variable with infinitely many possible values.

238. The mean of a Poisson distribution, where  is the average number of successes occurring in a specified interval, is .

239. The number of accidents that occur at a busy intersection in one month is an example of a Poisson random variable.

240. The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

241. The number of customers making a purchase out of 30 randomly selected customers has a Poisson distribution.

242. The largest value that a Poisson random variable X can have is n.

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.

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ANS: T PTS: 1 REF: SECTION 7.5 NAT: Analytic; Probability Distributions

244. In a Poisson distribution, the variance and standard deviation are equal.

245. In a Poisson distribution, the mean and standard deviation are equal.

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.

ANS: A PTS: 1 REF: SECTION 7.5

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.

ANS: B PTS: 1 REF: SECTION 7.5

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.

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. 1.3416 c. 0.1653 d. 6.05

ANS: C PTS: 1 REF: SECTION 7.5

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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.

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.

252. A community college has 150 word processors. The probability that any one of them will require repair on a given day is 0.025. 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.

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.

ANS: independent

PTS: 1 REF: SECTION 7.5

254. In a(n) ____________________ experiment, the probability of a success in an interval is the same for all equal-sized intervals.

ANS: Poisson

255. In a Poisson experiment, the probability of a success in an interval is ____________________ to the size of the interval.

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PTS: 1 REF: SECTION 7.5 NAT: Analytic; Probability Distributions

256. In Poisson experiment, the probability of more than one success in an interval approaches ____________________ as the interval becomes smaller.

ANS:

zero 0

257. A Poisson random variable is the number of successes that occur in a period of

____________________ or an interval of ____________________ in a Poisson experiment.

ANS: 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.

ANS:

mean

expected value

259. In the Poisson distribution, the mean is equal to the ____________________.

ANS: variance

260. In the Poisson distribution, the ____________________ is equal to the variance.

ANS: mean

261. The possible values of a Poisson random variable start at ____________________.

ANS:

zero 0

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262. A Poisson random variable is a(n) ____________________ random variable.

ANS: 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 ANS:

a. 0.2138 b. 0.4060 c. 0.8009

264. 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) ANS:

a. 0.847 b. 0.103 c. 0.715 d. 0.511

265. 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) ANS:

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b. 0.058 c. 0.986 d. 0.390

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.

ANS:

 = 5(30/60) = 2.5; P(X = 2) = 0.2565

267. {911 Phone Calls Narrative} Find the probability of receiving exactly eight calls in 15 minutes.

ANS:

 = 15(30/60) = 7.5; P(X = 8) = 0.1373

268. {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?

ANS:

 = 4(30/60) = 2.0; P(X = 0) = 0.1353

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.

ANS:

 = 10; P(X = 13) = 0.072

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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.

ANS:

 = 5; P(X = 7) = 0.105

271. {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.

ANS:

 = 10; P(X  5) = 0.971

NAT: Analytic; Probability Distributions 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.

ANS:

 =12; P(X  10) = 0.758

273. {Post Office Narrative} Find the probability that the number of arrivals between 3:30 and 4:00 P.M. is at least 10.

ANS:

 = 3; P(X  10) = 0.001

274. {{Post Office Narrative} Find the probability that the number of arrivals between 4:00 and 5:00 P.M.

is exactly two.

ANS:

 = 6; P(X = 2) = 0.045

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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?

ANS:

0.0504

NAT: Analytic; Probability Distributions 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.

ANS:

0.0892

277. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be at least 3 incidents in a year.

ANS:

0.9380

278. {Unsafe Levels of Radioactiviy Narrative} Find the probability that there will be at least 1 incident in a year.

ANS:

0.9975

279. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be no more than 1 incident in a year.

ANS:

0.0174

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PTS: 1 REF: SECTION 7.5 NAT: Analytic; Probability Distributions

280. {Unsafe Levels of Radioactivity Narrative} Find the variance of the number of incidents in one year.

ANS:

6

281. {Unsafe Levels of Radioactivity Narrative} Find the standard deviation of the number of incidents is in one year.

ANS:

2.45