Exam 5: Several Useful Discrete Distributions

The hypergeometric probability distribution formula calculates the probability of x successes 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, and N - M units have the characteristic that denotes failure.

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

The binomial distribution formula:

In an effort to persuade customers to reduce their electricity consumption, special discounts have been offered for low consumption customers. In Florida, the EPA estimates that 70% of the residences have qualified for these discounts. If we randomly select four residences in Florida, what is the probability exactly two of the four will have qualified for the discounts? ______________ If, instead, a sample of 20 residences is selected, what is the probability 12 or more will have qualified for the discounts? ______________

Let the random variable x have the Poisson distribution with mean 3. What is the probability x will fall in the interval ? ______________

An automobile service center can take care of 8 cars per day. Assume that the cars arrive at the service center randomly and independently of each other at a rate of 6 per hour, on average. What is the standard deviation of the number of cars that arrive at the center? ______________ What is the probability of the service center being empty in any given hour? ______________ What is the probability that exactly 6 cars will be in the service center at any point during a given hour? ______________ What is the probability that less than 2 cars will be in the service center at any point during a given hour? ______________

Students arrive at a health center, according to a Poisson distribution, at a rate of 4 every 15 minutes. Let x represent number of students arriving in a 15 minute time period. What is the probability that no more than 3 students arrive in a 15 minute time period? ______________ What is the probability that exactly 5 students arrive in a 15 minute time period? ______________ What is the probability that more than 5 students arrive in a 15-minute time period? ______________ What is the probability that between 4 and 8 students, inclusively, arrive in a 15-minute time period? ______________

Rebuilt ignition systems leave an aircraft repair shop at an average rate of 3 per hour. The assembly line needs four ignition systems in the next hour. What is the probability they will be available? ______________

An applicant must score at least 80 points on a particular psychological test to be eligible for the Peace Corps. From nearly 30 years experience, it is known 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. Find the mean of x. ______________ Find the standard deviation of x. ______________ Using the Empirical Rule, within what limits would you expect most (say, 95%) of the measurements to be? ________________________________________________________

The expected value, E(X), of a binomial probability distribution with n trials and probability p of success is:

An eight-cylinder automobile engine has two misfiring spark plugs. The mechanic removes all four plugs from one side of the engine. What is the probability the two misfiring spark plugs are among those removed? ______________ What is the mean number of misfiring spark plugs? ______________ What is the variance of the number of misfiring spark plugs? ______________

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the expected number of defects in the sample is 6.

A telephone survey of American families is conducted to determine the number of children in the average American family. Past experience has shown that 30% of the families who are telephoned will refuse to respond to the survey. Which of the following statements is not a binomial random variable?

The probability 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? ______________

Given that n is the number of trials and p is the probability of success in any one trial of a random experiment, the expected value of a binomial random variable equals:

The local Ford dealership claims 40% of their customers are return buyers, i.e., the customers previously purchased a Ford vehicle. Let the random variable x be the number of return buyers in a random sample of 10 recent customers. Find = ______________ Find P(x > 6) = ______________ What would you conclude about the accuracy of the dealership's claim if you found 9 repeat buyers in the sample? Explain your reasoning. ________________________________________________________

The number of teleport inquiries x in a timesharing computer system averages 0.2 per millisecond and follows a Poisson distribution. Find the probability no inquiries are made during the next millisecond. ______________ Find the probability no inquiries are made during the next 3 milliseconds. ______________

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

Let x be a binomial random variable with n = 15 and p = 0.20. Calculate P(x 4) using the binomial formula. ______________ Calculate P(x 4) using the cumulative binomial probabilities table in Appendix I of your book. ______________ Are the results of your two previous answers very different, or very similar? ______________ Calculate the mean of the random variable x. ______________ Calculate the standard deviation of the random variable x. ______________ Fill in the table below: Are the results consistent with Tchebysheff's Theorem? ______________ With the Empirical Rule? ______________

The binomial probability distribution is an example of discrete probability distributions.