Exam 5: Continuous Random Variables

The exponential distribution is sometimes called the waiting-time distribution, because it is used to describe the length of time between occurrences of random events.

Suppose x is a uniform random variable with c = 20 and d = 60. Find P(x > 44).

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 90. Find P(x < 42).

For a continuous probability distribution, the probability that x is between a and b is the same regardless of whether or not you include the endpoints, a and b, of the interval.

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?

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the probability that a randomly selected location in this village had between 17 and 18 inches of snow. Round to the nearest ten-thousandth.

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean θ = 8.5 minutes. If a customer just arrived, find the probability that the next customer will not arrive for at least 20 minutes.

A machine is set to pump cleanser into a process at the rate of 10 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 10.0 to 13.0 gallons per minute. Would you expect the machine to pump more than 12.85 gallons per minute?

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 2900 miles. What warranty should the company use if they want 96% of the tires to outlast the warranty?

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 60).

The total area under a probability distribution equals 1.

Transportation officials tell us that 80% of drivers wear seat belts while driving. Find the probability that more than 576 drivers in a sample of 700 drivers wear seat belts.

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 5. Find the value of a that makes the following probability statement true: P(x ≤ a) = 0.75.

The price of a gallon of milk follows a normal distribution with a mean of $3.20 and a standard deviation of $0.10. Find the price for which 12.3% of milk vendors exceeded.

Suppose that x has an exponential distribution with θ = 1.75. Find each probability. a. P(x ≥ 2) b. P(x < 2)

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Can the normal approximation be used to estimate the binomial distribution in this problem?

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 75°F to 95°F. What is the probability that a random day in August has a high temperature that exceeds 80°F?

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.

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door exceeds 50 years old. Round to the nearest ten-thousandth.

Assume that x is a binomial random variable with n = 100 and p = 0.60. Use a normal approximation to find P(x ≤ 65).