Exam 5: Several Useful Discrete Distributions

A quiz consists of 15 multiple choice questions. Each question has 5 choices, with exactly one correct choice. A student, totally unprepared for the quiz, guesses on each of the 15 questions. How many questions should the student expect to answer correctly? ______________ What is the standard deviation of the number of questions answered correctly? ______________ If at least 9 questions must be answered correctly to pass the quiz, what is the chance the student passes? ______________

Which of the following is a characteristic of a binomial problem?

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that one of the four parts is from company B is approximately .1099.

Which of the following statements regarding a binomial experiment is false, where n is the number of trials, and p is the probability of success in each trial?

Which of the following is not a characteristic of a binomial experiment?

Consider an experiment with 25 trials where the probability of success on any trial is 0.01, and let the random variable x be the number of successes among the 25 trials. Using the Poisson approximation to the binomial, what are: p(0) = ______________ p(1) = ______________ p(2) = ______________ p(3) = ______________

A new surgical procedure is said to be successful 90% of the time. Suppose the operation is performed five times and the results are assumed to be independent of one another. Find the probabilities listed below: Define x to be the number of successful operations. Then p = P(success) = 0.9 and n = 5. We will use the binomial formula. However, you may also use the cumulative binomial probabilities table in Appendix I of your book to calculate the necessary probabilities. All five operations are successful. ______________ Exactly four are successful. ______________ Less than two are successful. ______________

It is known that between 8 and 10 a.m. on Saturdays, cars arrive at a toll station in Indiana at a rate of 60 per hour. Assume that a Poisson process is occurring, and that the random variable x represents the number of cars arriving at the station between 9:00 and 9:05 a.m. What is the expected number of cars arriving at the toll station between 9:00 and 9:05 a.m.? ______________ What is the standard deviation of the number of cars arriving at the toll station between 9:00 and 9:05 a.m.? ______________ Find P(x = 0). ______________ Find P(x = 2). ______________ Find P(x = 5). ______________ Find P(x = 10). ______________

Let x be a binomial random variable with n = 20 and p = 0.05. Calculate using the cumulative binomial probabilities table in Appendix I of your book. ______________ Use the Poisson approximation to calculate . ______________ Compare your results. Is the approximation accurate? ______________

The Poisson probability distribution is an example of continuous probability distribution.

A Poisson process is the occurrence of a series of events of a given type in a random pattern over time or space such that (1) the number of occurrences within a specified time or space can equal any integer between zero and infinity, (2) the number of occurrences within one unit of time or space is independent of that in any other such (non-overlapping) unit, and (3) the probability of occurrences is the same in all such units.

To determine the effectiveness of a diet to reduce the amount of serum cholesterol, 20 people were randomly selected to try the diet. After they had been on the diet a sufficient amount of time, their blood was tested to determine the level of serum cholesterol. The clinic running the experiment will approve and endorse the diet only if at least 16 of the 20 people lower their cholesterol significantly. What is the random variable of interest? ________________________________________________________ What is the probability the clinic will erroneously endorse the diet when, in fact, it is useless in treating high serum cholesterol?

A student has decided to rent three movies for a three-day weekend. If there are 4 action movies and 6 romance movies that are of equal interest to the student, what is the probability that the student will select 1 romance movie and 2 action movies? ______________

The binomial distribution is used to describe continuous random variables.

Three yellow and two blue pencils are in a drawer. If we randomly select two pencils from the drawer, find the probability distribution of x, the number of yellow pencils selected.

Which of the following correctly describes a Poisson random variable?

When sampling without replacement, the appropriate probability distribution is:

Which probability distribution is appropriate when the events of interest occur randomly, independently of one another, and rarely?

The Poisson probability tables list the probabilities of x occurrences in a Poisson process for various values of , the mean number of occurrences.

The hypergeometric probability distribution: