Exam 5: Several Useful Discrete Distributions

Intensive Care Unit Narrative The number x of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to four persons per day. -Refer to Intensive Care Unit Narrative. What is the probability that the number of people entering the intensive care unit on a particular day is less than or equal to two?

Hotels, like airlines, often overbook, counting on the fact that some people with reservations will cancel at the last minute. A certain hotel chain has found that 20% of the reservations will not be used. a. If we randomly selected 15 reservations, what is the probability more than 8 but fewer than 12 reservations will be used? b. If four reservations are made, what is the chance fewer than two will cancel?

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. What are p(0), p(1), p(2), and p(3), based on the Poisson approximation to the binomial?

From a group of ten bank officers, three are selected at random to be relocated and to supervise new branch offices. If two of the ten officers are women and eight are men, what is the probability exactly one of the officers to be relocated will be a woman?

The Poisson distribution is appropriate to determine the probability of a given number of defective items in a shipment.

In a binomial experiment, the probability of success is the same on every trial.

Health Care Narrative Students arrive at a health centre, according to a Poisson distribution, at a rate of four every 15 minutes. Let x represent number of students arriving in a 15-minute period. -Refer to Health Care Narrative. What is the probability that exactly five students arrive in a 15-minute period?

Scholarship Narrative A college has seven applicants for three scholarships: four females and three males. Suppose the seven applicants are equally qualified and no preference is given by the selection committee for choosing either gender. Let x equal the number of female students chosen for the three scholarships. -Refer to Scholarship Narrative. What is the probability that none of the three males will receive a scholarship?

A binomial experiment requires that the success and failure probabilities be constant from one trial to the next, and also that these two probabilities be equal to each other.

A fellow student tells you he is working with a discrete random variable, x, which takes on integer values from 0 to 50 and has a mean of 40 and a variance of 10. Is x a binomial random variable?

A binomial probability distribution shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.

The distribution of the number of phone calls to a doctor's office in a one-hour time period is likely to be described by a binomial distribution.

Computer Disks Narrative The quality of computer disks is measured by sending the disks through a certifier that counts the number of missing pulses. A certain brand of computer disks averages 0.1 missing pulses per disk. Let the random variable x denote the number of missing pulses. -Refer to Computer Disks Narrative. Find the probability the next disk inspected will have more than one missing pulse.

Which of the following CANNOT generate a Poisson distribution?

A binomial experiment is a sequence of n identical trials such that both of the following properties hold: a.) each trial produces one of two outcomes that are conventionally called "success" and "failure," and b.) each trial is independent of any other trial so that the probability of success or failure is constant from trial to trial.

The binomial probability distribution could be used to describe the speed of tennis balls when the players are serving.

Scholarship Narrative A college has seven applicants for three scholarships: four females and three males. Suppose the seven applicants are equally qualified and no preference is given by the selection committee for choosing either gender. Let x equal the number of female students chosen for the three scholarships. -Refer to Scholarship Narrative. Write a formula for p(x), the probability distribution of x.

Computer Disks Narrative The quality of computer disks is measured by sending the disks through a certifier that counts the number of missing pulses. A certain brand of computer disks averages 0.1 missing pulses per disk. Let the random variable x denote the number of missing pulses. -Refer to Computer Disks Narrative. What is the distribution of x?

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.

An applicant must score at least 80 points on a particular psychological test to be eligible to join CUSO (Canadian University Services Overseas). From several years' experience, it is known that 60% of the applicants meet this requirement. Let the random variable x be the number of applicants who meet this requirement out of a (randomly selected) group of 25 applicants. a. Find the mean of x. b. Find the standard deviation of x. c. Using the Empirical Rule, within what limits would you expect most (say, 95%) of the measurements to be?