STAT 400 Lecture AL1 1 Practice Problems 2 Part 2 Spring 2015 Dalpiaz Let X be a discrete random variable with p m f 3 k p k c 4 k 5 6 7 8 a Find the value of c that makes this is a valid probability distribution b Find P X is even c Find the moment generating function of X M X t For which values of t does it exist d Find E X 2 Suppose a discrete random variable X has the following probability distribution P X k ln 2 k k k 1 2 3 a Verify that this is a valid probability distribution b Find X E X by finding the sum of the infinite series c Find the moment generating function of X M X t d Use M X t to find X E X e Find X E X by comparing it to the expected value of a Poisson random variable with mean ln 2 Hint f The answers to b d and e should be the same Find X2 Var X 3 The manufacturer of a price reading scanner claims that the probability that the scanner will misread a price is 0 01 Shortly after one of the scanners was installed in a supermarket the store manager tested the performance of the scanner Assume that the outcome of each scan is independent of the others a Assuming that their claim is correct what is the probability that the first time the scanner misreads a price will be on the seventh scan b Find the probability that the second error will be on the twenty fifth scan c Find the probability that the third error will be on the twenty fifth scan d Suppose 25 prices were read Find the probability of more than one error e Suppose 25 prices were read Find the probability of exactly one error f Suppose 25 prices were read Find the probability of exactly two errors 4 A purchasing agent is considering an acceptance plan for incoming lots of some manufactured product The plan calls for taking a random sample of 20 items with replacement from each lot If there is at most one defective in the sample the lot is accepted otherwise the lot is rejected Find the probability of accepting the lot if the defective rate is a 1 b 5 c 10 5 A credit card company sends a special pre approved low interest application form to a random sample of 25 individuals Past experience indicates that about 10 of the people who receive such an application eventually reply Let X denote the number of replies received in this sample of 25 individuals a Find the probability of receiving at most 3 replies b Find the probability of receiving exactly 3 replies c Find the probability of receiving at least 3 replies d Find the probability of receiving between 2 and 4 both inclusive replies 6 The number of tornadoes observed in a particular region during a 1 year period is random and has a Poisson distribution mean of 8 tornadoes a Find the probability that less than 4 tornadoes will be observed during a 1 year period b What is the probability of observing exactly 10 tornadoes during a 1 year period c What is the probability of observing at most 2 tornadoes during a 6 month period d What is the probability of observing at least 2 tornadoes during a 3 month period 7 The marketing manager of a company has noted that she usually receives an average of 10 complaint calls from customers during a week 5 working days and that the calls occur at random according to a Poisson distribution a Find the probability of her receiving exactly 3 such calls in a single day b Find the probability of her receiving no complaint calls in a single day c What is the probability that on 2 out of 5 days there will be no complaint calls d What is the probability that the first such call will occur during the first half of the third day e Find the probability that she receives at least 5 complaint calls over two days 8 One in 5 000 salmon caught in Alaska s Bristol Bay has parasites that make it unfit for human consumption Use the Poisson approximation to find the probability that out of a shipment of 1 800 fish 2 or more will have to be destroyed due to parasites 9 11 9 10 Alex sells Exciting World of Statistics videos over the phone to earn some extra cash during the economic crisis Only 10 of all calls result in a sale Assume that the outcome of each call is independent of the others a What is the probability that Alex makes his first sale on the fifth call b What is the probability that Alex makes his first sale on an odd numbered call c What is the probability that it takes Alex at least 10 calls to make his first sale d What is the probability that it takes Alex at most 6 calls to make his first sale e What is the probability that Alex makes his second sale on the ninth call f What is the probability that Alex makes his second sale on an odd numbered call Hint 11 Consider Answer 0 9 2 Answer On one side you will have 0 19 Answer On the other side you will have a geometric series g What is the probability that Alex makes his third sale on the 13 th call h If Alex makes 15 calls what is the probability that he makes exactly 3 sales i If Alex makes 15 calls what is the probability that he makes at least 2 sales j If Alex makes 15 calls what is the probability that he makes at most 2 sales 12 13 a Let X have a Poisson distribution with variance of 3 Find P X 2 b If X has a Poisson distribution such that 3 P X 1 P X 2 find P X 4 Suppose the number of air bubbles in window glass has a Poisson distribution with an average of 0 3 air bubbles per square foot In a 4 by 3 window find the probability that there are a exactly 5 air bubbles From the textbook Ninth Eighth Seventh 2 4 10 2 4 12 2 4 14 2 4 15 2 4 18 2 4 20 2 5 3 2 5 10 2 5 10 2 6 8 2 6 8 2 6 8 b at least 5 air bubbles 1 Let X be a discrete random variable with p m f k 3 p k c 4 a k 5 6 7 8 Find the value of c that makes this is a valid probability distribution Must have p x 1 k 5 4 all x 5 3 3 k first term 4 4 1 base 3 k 5 1 4 k 3 k 3 c c 1 243 1024 1 4 k 5 4 243 256 OR 3 4 k 5 k 3 …
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