Exam 4: Discrete Random Variables

Classify the following random variable according to whether it is discrete or continuous. The number of pills in a container of vitamins

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Find the probability that exactly two babies are born during a randomly selected hour.

The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 5 goals per game. Find the probability that the team scored exactly three goals in each of four randomly selected games.

Suppose a Poisson probability distribution with ? = 9.6 provides a good approximation of the distribution of a random variable x. Find ? for x.

Given that x is a hypergeometric random variable with N = 10, n = 3, and r = 6, compute P(x = 0).

Explain why the following is or is not a valid probability distribution for the discrete random variable x. x 10 20 30 40 50 p(x) .3 .2 .2 .2 .2

An airline has requests for standby flights at half of the usual one-way air fare. Past experience has shown that these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, the only other choice is to fly first class on the next flight out. Suppose that the usual one-way air fare to a certain city is $156 and the cost of flying first class is $355. Should a passenger who wishes to fly to this city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying standby.]

Given that x is a hypergeometric random variable with N = 9, n = 3, and r = 5, compute the standard deviation of x.

Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the mean of this distribution.

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 7.1. Find the probability of observing exactly four accidents on this stretch of road next month.

Calculate the mean for the discrete probability distribution shown here. X 1 4 8 11 P(X) 0.28 0.24 0.17 0.31

Consider the given discrete probability distribution. Find the probability that x equals 5. x 3 5 7 9 P(x) 0.33 ? 0.27 0.31

The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 7.4. Find the probability that fewer than six tickets are written on a randomly selected day.

According to a recent study, 1 in every 6 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women sampled have not been the victim of domestic abuse.

Given that x is a hypergeometric random variable with N = 8, n = 4, and r = 3, compute the variance of x.

The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 7.5. Interpret the value of the mean.

Suppose x is a random variable for which a Poisson probability distribution with λ = 3 provides a good approximation. a. Graph p(x) for x = 0, 1, 2, 3, 4, 5, 6. b. Find μ and σ for x. c. What is the probability that x will fall in the interval μ ± σ?

A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution below: Price \ 10.00 \ 12.50 \ 15.00 Probability 0.25 0.45 0.3 Suppose the mean cost of the frogs is $12.63 per week. Interpret this value.

Which binomial probability is represented on the screen below?

Given that x is a hypergeometric random variable with N = 10, n = 5, and r = 6, find each probability. a. P(x = 0) b. P(x = 1) c. P(x ≤ 1) d. P(x ≥ 2)