Exam 5: Several Useful Discrete Distributions

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

The mean of a Poisson distribution, where is the average number of successes occurring in a specified interval, is .

In a book, 2 misprints occur per 100 pages. Using the cumulative Poisson probability table available in your text, we can determine which of the following probabilities in a book of 500 pages?

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 2 sales is less than 0.10.

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

A company has five applicants for two positions: three women and two men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let x equal the number of men chosen to fill the two positions. What is the mean of the probability distribution of x? ______________ What is the variance of the probability distribution of x? ______________ What is the standard deviation of the probability distribution of x? ______________

Four in ten Americans who travel by car look for gas and food outlets that are close to or visible from the highway. Suppose a random sample of n = 20 Americans who travel by car are asked how they determine where to stop for food and gas. Let x be the number in the sample who respond that they look for gas and food outlets that are close to or visible from the highway. Calculate the mean of x. ______________ Calculate the variance of x. ______________ Calculate the interval . ______________ What values of the binomial random variable x fall into this interval? ______________ Find . ______________ How does this compare with the fraction in the interval for any distribution? For mound-shaped distributions? ________________________________________________________

The number of traffic accidents per day on a certain section of highway is thought to be Poisson distributed with a mean equal 2.19. Then the standard deviation of number of accidents is:

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.

Using the hypergeometric formula, :

Given a Poisson random variable x, where the average number of times an event occurs in a certain period of time is 2.5, then P(x = 0) is:

From past experience, it is known 90% of one-year-old children can distinguish their mother's voice from the voice of a similar sounding female. A random sample of 20 one-year-olds are given this voice recognition test. Find the probability at least 3 children do not recognize their mother's voice. ______________ Find the probability all 20 children recognize their mother's voice. ______________ Let the random variable x denote the number of children who do not recognize their mother's voice. Find the mean of x. ______________ Let the random variable x denote the number of children who do not recognize their mother's voice. Find the variance of x. ______________ Find the probability that at most 4 children do not recognize their mother's voice. ______________

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

A manufacturer of golf balls uses a production process that produces 10 percent defective balls. A quality inspector takes samples of a week's output with replacement. Using the cumulative binomial probability table available in your text, the inspector can determine which of the following probabilities?

It has been alleged that 40 percent of all college students favor Dell computers. If this were true, and we took a random sample of 50 students, the binomial probability table for cumulative values of x available in your text, would reveal which of the following probabilities?

In 1996, the average combined SAT score (math + verbal) for students in the United States was 1013, and 41% of all high school graduates took this test. Suppose that 500 students are randomly selected throughout the United States. Which of the following random variables has an approximate binomial distribution? The number of students who took the SAT. ____________________________ The scores of the 500 students on the SAT. ____________________________ The number of students who scored above average on the SAT. ____________________________ The amount of time it took each student to complete the SAT. ____________________________

Hypergeometric probability distributions is an example of discrete probability distributions.

As a rule of thumb, if the sample size n is large relative to the population size N in particular, if n / N 0.05 then the resulting experiment will not be binomial.

A small community college in Ohio 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?

An oil firm plans to drill 20 wells, each having probability 0.2 of striking oil. Each well costs $20,000 to drill; a well which strikes oil will bring in $750,000 in revenue. Find the expected gain from the 20 wells. ______________