Exam 4: Discrete Random Variables

The expected value of a discrete random variable must be one of the values in which the random variable can result.

It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Use a binomial probability table to find the probability that exactly 5 of the students reported that they did have at least one tattoo.

A dice game involves rolling three dice and betting on one of the six numbers that are on the dice. The game costs $11 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: 0 -$11 125/216 Number of dice with your number Profit Probability 0 -\ 11 125/216 1 \ 11 75/216 2 \ 13 15/216 3 \ 33 1/216 Find your expected profit from playing this game.

If x is a binomial random variable, compute p(x) for n = 5, x = 1, p = 0.4.

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 random variable x represents the number of boys in a family with three children. Assuming that births of boys and girls are equally likely, find the mean and standard deviation for the random variable x.

Consider the given discrete probability distribution. Find P(x > 3). x 1 2 3 4 5 p(x) .1 .2 .2 .3 .2

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 6. Find the probability that exactly four road construction projects are currently taking place in this city.

Consider the given discrete probability distribution. x 1 2 3 4 5 p(x) .1 .2 .2 .3 .2 a. Find $\mu = E ( x )$ . b. Find $\sigma = \sqrt { E \left[ ( x - \mu ) ^ { 2 } \right] }$ . c. Find the probability that the value of $x$ falls within one standard deviation of the mean. Compare this result to the Empirical Rule.

Consider the given discrete probability distribution. Find P(x < 2 or x > 3). x 1 2 3 4 5 p(x) .1 .2 .2 .3 .2

We believe that 90% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 24 students from the population and observed fewer than five in our sample who consider statistics to be an exciting subject. Make an inference about the belief that 90% of the students consider statistics to be an exciting subject.

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

Explain why the following is or is not a valid probability distribution for the discrete random variable x. x 1 0 1 2 3 p(x) .1 .2 .3 .3 .1

Given that x is a hypergeometric random variable with N = 8, n = 4, and r = 3: a. Display the probability distribution in tabular form. b. Find P(x ≤ 2).

The hypergeometric random variable x counts the number of successes in the draw of n elements from a set of N elements containing r successes.

Suppose that 4 out of 12 liver transplants done at a hospital will fail within a year. Consider a random sample of 3 of these 12 patients. What is the probability that all 3 patients will result in failed transplants?

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

If x is a binomial random variable, calculate σ2 for n = 70 and p = 0.2.

Explain why the following is or is not a valid probability distribution for the discrete random variable x. x 0 2 4 6 8 p(x) -.1 .1 .2 .3 .5

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 7. Find the probability that more than four road construction projects are currently taking place in the city.