Exam 4: Discrete Random Variables

A dice game involves rolling three dice and betting on one of the six numbers that are on the dice. The game costs $12 to play, and you win if the number you bet appears on any of the dice. The Distribution for the outcomes of the game (including the profit)is shown below: Number of dice with your number Profit Probability 0 -\ 12 125/216 1 \ 12 75/216 2 \ 14 15/216 3 \ 36 1/216 Find your expected profit from playing this game.

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

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 $150 and the cost of flying first class is $315. 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.]

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

Suppose a Poisson probability distribution with $\lambda = 6.3$ provides a good approximation of the distribution of a random variable $x$ . Find $\sigma$ for $x$ .

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.5. Find the probability that fewer than two accidents will occur on this stretch of road during a month.

An alarm company reports that the number of alarms sent to their monitoring center from customers owning their system follow a Poisson distribution with λ = 4.6 alarms per year. Find the Probability that a randomly selected customer had more than 7 alarms reported.

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 3. Find the probability that more than four road construction Projects are currently taking place in the city.

Find the mean and standard deviation of the probability distribution for the random variable x, which represents the number of cars per household in a small town. x P(x) 0 .125 1 .428 2 .256 3 .108 4 .083

Solve the problem. Round to four decimal places. -A literature professor decides to give a 10-question true-false quiz. She wants to choose the passing grade such that the probability of passing a student who guesses on every question is less Than .10. What score should be set as the lowest passing grade?

Given that x is a hypergeometric random variable, compute p(x)for N = 8, n = 5, r = 3, and x = 2.

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 45% of all employees believe their company president Possesses low ethical standards. Suppose 20 of a companyʹs employees are randomly and Independently sampled and asked if they believe their company president has low ethical Standards and their years of experience at the company. Could the probability distribution for the Number of years of experience be modelled by a binomial probability distribution?

The number of homeruns hit during a major league baseball game follows a Poisson distribution with $\lambda = 3.2$ . Find the mean and standard deviation for this distribution.

If $x$ is a binomial random variable, calculate $\mu$ for $n = 100$ and $p = 0.8$ .

For a binomial distribution, if the probability of success is .63 on the first trial, what is the probability of success on the second trial?

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the standard deviation of the distribution. x 2 3 5 8 10 p(x) 0.10 0.20 0.30 0.30 0.10

Classify the following random variable according to whether it is discrete or continuous. The blood pressures of a group of students the day before the final exam

Compute $\left( \begin{array} { l } 5 \\ 0 \end{array} \right)$ .

Given that x is a hypergeometric random variable with N = 10, n = 5, and r = 6, compute the mean of x.

Consider the given discrete probability distribution. Find P(x=1 or x=2) . x 0 1 2 3 4 5 p(x) .30 .25 .20 .15 .05 .05