Exam 5: Several Useful Discrete Distributions

A small community college in Ontario has four student organizations (A, B, C, and D). Organization A has 5 students, B has 8, C has 10, and D has 12. It is thought that new students have no preference for one of these organizations over the other. If seven new students are admitted to the college, what is the probability that one student will choose organization A, one will choose B, two will choose C, and three will choose D?

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

What is the expected value, E(X), of a binomial probability distribution with n trials and a probability p of success?

The probability that the 1993-94 flu vaccine immunizes those receiving it is 0.97. If a random sample of 200 people receive the vaccine, what is the probability the vaccine will be ineffective on at most 5 people?

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 more than five students arrive in a 15-minute period?

In which of the following sampling methods would the hypergeometric probability distribution be used instead of the binomial distribution?

If the standard deviation for a Poisson random variable is known to be 3.60, what is its expected value?

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 only one female will receive a scholarship?

Automobile Service Centre Narrative An automobile service centre can take care of eight cars per hour. Assume that the cars arrive at the service centre randomly and independently of each other at a rate of six per hour, on average. -Refer to Automobile Service Centre Narrative. What is the probability of the service centre being empty in any given hour?

A life insurance salesperson makes 15 sales calls daily. The chance of making a sale on each call is 0.40. The probability that he will make at most two sales is less than 0.10.

A binomial random variable is an example of a discrete random variable.

The binomial distribution is used to describe continuous random variables.

Let x be a Poisson random variable with mean 3. What is the probability that x will fall in the interval $\mu$ ± 2 $\sigma$ ?

The Poisson random variable is the number of successes achieved when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success.

A hypergeometric probability distribution shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling with replacement, and the probability of success, therefore, changes from one trial to the next.

In a book, two misprints occur per 100 pages. Using the cumulative Poisson probability table available in your text, which of the following probabilities can we determine for a book of 500 pages?

A manufacturer of tennis balls uses a production process that produces 5% defective balls. A quality inspector takes samples of a week's output, with replacement. Using the cumulative binomial probability table available in your text, which of the following probabilities can the inspector determine?

Which of the following best describes a Poisson random variable?

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 (nonoverlapping) unit, and (3) the probability of occurrences is the same in all such units.

Defective Items Narrative A random sample of 4 units is taken from a group of 15 items in which 4 units are known to be defective. Assume that sampling occurs without replacement, and the random variable x represents the number of defective units found in the sample. -Refer to Defective Items Narrative. What kind of discrete random variable x is this?