Continuation of Exercise 3-109. (a) What is the probability that you must call six times

3-97. Suppose X has a Poisson distribution with a mean of 4. Determine the following probabilities:

Suppose X has a Poisson distribution with a mean of 0.4. Determine the following probabilities:

3-99. Suppose that the number of customers that enter a bank in an hour is a Poisson random variable, and suppose that Determine the mean and variance of X

The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour. (a) What is the probability that there are exactly 5 calls in one hour? (b) What is the probability that there are 3 or less calls in one hour? (c) What is the probability that there are exactly 15 calls in two hours? (d) What is the probability that there are exactly 5 calls in 30 minutes?

3-101. The number of flaws in bolts of cloth in textile manufacturing is assumed to be Poisson distributed with a mean of 0.1 flaw per square meter. (a) What is the probability that there are two flaws in 1 square meter of cloth? (b) What is the probability that there is one flaw in 10 square meters of cloth?

When a computer disk manufacturer tests a disk, it writes to the disk and then tests it using a certifier. The certifier counts the number of missing pulses or errors. The number of errors on a test area on a disk has a Poisson distribution with 0.2. (a) What is the expected number of errors per test area? (b) What percentage of test areas have two or fewer errors?

3-103. The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile. (a) What is the probability that there are no cracks that require repair in 5 miles of highway? (b) What is the probability that at least one crack requires repair in mile of highway? (c) If the number of cracks is related to the vehicle load on the highway and some sections of the highway have a heavy load of vehicles whereas other sections carry a light load, how do you feel about the assumption of a Poisson distribution for the number of cracks that require repair?

The number of failures for a cytogenics machine from contamination in biological samples is a Poisson random variable with a mean of 0.01 per 100 samples. (a) If the lab usually processes 500 samples per day, what is the expected number of failures per day?

3-105. The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.05 flaw per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel. (a) What is the probability that there are no surface flaws in an autos interior? (b) If 10 cars are sold to a rental company, what is the probability that none of the 10 cars has any surface flaws? (c) If 10 cars are sold to a rental company, what is the probability that at most one car has any surface flaws?

The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failure per hour. (a) What is the probability that the instrument does not fail in an 8-hour shift? (b) What is the probability of at least one failure in a 24-hour day?

3-107. A shipment of chemicals arrives in 15 totes. Three of the totes are selected at random, without replacement, for an inspection of purity. If two of the totes do not conform to purity requirements, what is the probability that at least one of the nonconforming totes is selected in the sample?

The probability that your call to a service line is answered in less than 30 seconds is 0.75. Assume that your calls are independent. (a) If you call 10 times, what is the probability that exactly 9 of your calls are answered within 30 seconds? (b) If you call 20 times, what is the probability that at least 16 calls are answered in less than 30 seconds? (c) If you call 20 times, what is the mean number of calls that are answered in less than 30 seconds?

3-109. Continuation of Exercise 3-108. (a) What is the probability that you must call four times to obtain the first answer in less than 30 seconds? (b) What is the mean number of calls until you are answered in less than 30 seconds?

Continuation of Exercise 3-109. (a) What is the probability that you must call six times in order for two of your calls to be answered in less than 30 seconds? (b) What is the mean number of calls to obtain two answers in less than 30 seconds?

3-111. The number of messages sent to a computer bulletin board is a Poisson random variable with a mean of 5 messages per hour. (a) What is the probability that 5 messages are received in 1 hour? (b) What is the probability that 10 messages are received in 1.5 hours? (c) What is the probability that less than two messages are received in one-half hour?

A Web site is operated by four identical computer servers. Only one is used to operate the site; the others are spares that can be activated in case the active server fails. The probability that a request to the Web site generates a failure in the active server is 0.0001. Assume that each request is an independent trial. What is the mean time until failure of all four computers?

3-113. The number of errors in a textbook follow a Poisson distribution with a mean of 0.01 error per page. What is the probability that there are three or less errors in 100 pages?

The probability that an individual recovers from an illness in a one-week time period without treatment is 0.1. Suppose that 20 independent individuals suffering from this illness are treated with a drug and 4 recover in a one-week time period. If the drug has no effect, what is the probability that 4 or more people recover in a one-week time period?

3-115. Patient response to a generic drug to control pain is scored on a 5-point scale, where a 5 indicates complete relief. Historically the distribution of scores is 1 23 4 5 0.05 0.1 0.2 0.25 0.4 Two patients, assumed to be independent, are each scored. (a) What is the probability mass function of the total score? (b) What is the probability mass function of the average score?

In a manufacturing process that laminates several ceramic layers, 1% of the assemblies are defective. Assume that the assemblies are independent. (a) What is the mean number of assemblies that need to be checked to obtain five defective assemblies? (b) What is the standard deviation of the number of assemblies that need to be checked to obtain five defective assemblies?

3-117. Continuation of Exercise 3-116. Determine the minimum number of assemblies that need to be checked so that the probability of at least one defective assembly exceeds 0.95.

Determine the constant c so that the following function is a probability mass function: for x 1, 2, 3, 4.

3-119. A manufacturer of a consumer electronics product expects 2% of units to fail during the warranty period. A sample of 500 independent units is tracked for warranty performance. (a) What is the probability that none fails during the warranty period? (b) What is the expected number of failures during the warranty period? (c) What is the probability that more than two units fail during the warranty period?

Messages that arrive at a service center for an information systems manufacturer have been classified on the basis of the number of keywords (used to help route messages) and the type of message, either email or voice. Also, 70% of the messages arrive via email and the rest are voice. number of keywords 01234 email 0.1 0.1 0.2 0.4 0.2 voice 0.3 0.4 0.2 0.1 0 Determine the probability mass function of the number of keywords in a message.

The random variable X has the following probability distribution: x 2358 probability 0.2 0.4 0.3 0.1 Determine the following:

Determine the probability mass function for the random variable with the following cumulative distribution function:

Each main bearing cap in an engine contains four bolts. The bolts are selected at random, without replacement, from a parts bin that contains 30 bolts from one supplier and 70 bolts from another. (a) What is the probability that a main bearing cap contains all bolts from the same supplier? (b) What is the probability that exactly three bolts are from the same supplier?

Assume the number of errors along a magnetic recording surface is a Poisson random variable with a mean of one error every bits. A sector of data consists of 4096 eight-bit bytes. (a) What is the probability of more than one error in a sector? (b) What is the mean number of sectors until an error is found?

3-125. An installation technician for a specialized communication system is dispatched to a city only when three or more orders have been placed. Suppose orders follow a Poisson distribution with a mean of 0.25 per week for a city with a population of 100,000 and suppose your city contains a population of 800,000. (a) What is the probability that a technician is required after a one-week period? (b) If you are the first one in the city to place an order, what is the probability that you have to wait more than two weeks from the time you place your order until a technician is dispatched?

From 500 customers, a major appliance manufacturer will randomly select a sample without replacement. The company estimates that 25% of the customers will provide useful data. If this estimate is correct, what is the probability mass function of the number of customers that will provide useful data? (a) Assume that the company samples 5 customers. (b) Assume that the company samples 10 customers.

3-127. It is suspected that some of the totes containing chemicals purchased from a supplier exceed the moisture content target. Samples from 30 totes are to be tested for moisture content. Assume that the totes are independent. Determine the proportion of totes from the supplier that must exceed the moisture content target so that the probability is 0.90 that at least one tote in the sample of 30 fails the test.

Messages arrive to a computer server according to a Poisson distribution with a mean rate of 10 per hour. Determine the length of an interval of time such that the probability that no messages arrive during this interval is 0.90.

3-129. Flaws occur in the interior of plastic used for automobiles according to a Poisson distribution with a mean of 0.02 flaw per panel. (a) If 50 panels are inspected, what is the probability that there are no flaws? (b) What is the expected number of panels that need to be inspected before a flaw is found? (c) If 50 panels are inspected, what is the probability that the number of panels that have one or more flaws is less than or equal to 2?

Derive the mean and variance of a hypergeometric random variable (difficult exercise).

Show that the function f(x) in Example 3-5 satisfies the properties of a probability mass function by summing the infinite series.

Derive the formula for the mean and standard deviation of a discrete uniform random variable over the range of integers .

A company performs inspection on shipments from suppliers in order to defect nonconforming products. Assume a lot contains 1000 items and 1% are nonconforming. What sample size is needed so that the probability of choosing at least one nonconforming item in the sample is at least 0.90? Assume the binomial approximation to the hypergeometric distribution is adequate.

A company performs inspection on shipments from suppliers in order to detect nonconforming products. The companys policy is to use a sample size that is always 10% of the lot size. Comment on the effectiveness of this policy as a general rule for all sizes of lots.

Surface flaws in automobile exterior panels follow a Poisson distribution with a mean of 0.1 flaw per panel. If 100 panels are checked, what is the probability that fewer than five panels have any flaws?

A large bakery can produce rolls in lots of either 0, 1000, 2000, or 3000 per day. The production cost per item is $0.10. The demand varies randomly according to the following distribution: demand for rolls 0 1000 2000 3000 probability of demand 0.3 0.2 0.3 0.2 Every roll for which there is a demand is sold for $0.30. Every roll for which there is no demand is sold in a secondary market for $0.05. How many rolls should the bakery produce each day to maximize the mean profit?

A manufacturer stocks components obtained from a supplier. Suppose that 2% of the components are defective and that the defective components occur independently. How many components must the manufacturer have in stock so that the probability that 100 orders can be completed without reordering components is at least 0.95?