Exam 4: Random Variables and Probability Distributions

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

Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 516 tomatoes and a standard deviation of 30 tomatoes. If there are 474 tomatoes available to be sold at the roadside stand at the beginning of a day, what is the probability that they will all be sold?

An online retailer reimburses a customer's shipping charges if the customer does not receive his order within one week. Delivery time (in days) is exponentially distributed with a mean of 3.2 days. What percentage of customers have their shipping charges reimbursed?

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

A small life insurance company has determined that on the average it receives 5 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.

The board of examiners that administers the real estate broker's examination in a certain state found that the mean score on the test was 487 and the standard deviation was 72. If the board wants to set the passing score so that only the best 10% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

Suppose that the random variable x has an exponential distribution with θ = 3. a. Find the probability that x assumes a value more than three standard deviations from µ. b. Find the probability that x assumes a value less than one standard deviation from µ. c. Find the probability that x assumes a value within a half standard deviation of µ.

For a binomial distribution, if the probability of success is .48 on the first trial, what is the probability of failure on the second trial?

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

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.

Determine if it is appropriate to use the normal distribution to approximate a binomial distribution when n = 36 and p = 0.5.

Mamma Temte bakes six pies each day at a cost of $2 each. On 11% of the days she sells only two pies. On 17% of the days, she sells 4 pies, and on the remaining 72% of the days, she sells all six pies. If Mama Temte sells her pies for $4 each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given away.]

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?

The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 2500 miles. What is the probability a certain tire of this brand will last between 54,750 miles and 55,500 miles?

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 30$ and $d = 90$ . Find $P ( 30 \leq x \leq 60 )$ .

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 diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 8.5 millimeters. What is the probability of a randomly selected ball bearing having a diameter less than 4.5 millimeters?

For a binomial distribution, which probability is not equal to the probability of 1 success in 5 trials where the probability of success is .4?

A machine is set to pump cleanser into a process at the rate of 6 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 5.5 to 6.5 gallons per minute. Find the probability that the machine pumps less than 5.75 gallons during a randomly selected minute.

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.