Exam 6: Continuous Probability Distributions

The cumulative distribution function F(x)of a continuous random variable X with the probability density function f(x)is which of the following?

Suppose Jennifer is waiting for a taxi cab.A taxi cab's arrival time is equally likely at any constant time range in the next 12 minutes. A)Calculate the expected arrival time. B)What is the probability that a taxi arrives in three minutes or less?

It is known that the length of a certain product X is normally distributed with μ = 20 inches and σ = 4 inches.How is the probability related to ?

Exhibit 6-3.Patients scheduled to see their primary care physician at a particular hospital wait,on average,an additional eight minutes after their appointment is scheduled to start.Assume the time that patients wait is exponentially distributed. Refer to Exhibit 6-3.What is the probability a randomly selected patient will have to wait more than 10 minutes?

Find the following probabilities for a standard normal random variable Z. A) b. c. d.

The weight of competition pumpkins at the Circleville Pumpkin Show in Circleville,Ohio,can be represented by a normal distribution with a mean of 703 pounds and a standard deviation of 347 pounds. A)Find the probability that a randomly selected pumpkin weighs at least 1622 pounds. B)Find the probability that a randomly selected pumpkin weighs between 465.1 and 1622 pounds.

Find the mean of the lognormal variable if the mean and standard deviation of the underlying normal variable are 2 and 0.8,respectively.

A producer has a history of making bad movies.The movies he has produced have averaged $160,000 dollars at the box office with a standard deviation of $185,000.He thinks his latest movie will be a huge hit.How much will the movie earn at the box office if it is only expected to earn that much or more 0.5% of the time? (Assume that the profit at the box office is normally distributed. )

Exhibit 6-5.The mean travel time to work is 25.2 minutes (U.S.Census 2010).Further,suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes. Refer to Exhibit 6-5.What is the probability a randomly selected U.S.worker has a commute time of more than half an hour?

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds.What is the probability a rider must wait between 1 minute and 1.5 minutes?

The probability density function of a continuous uniform distribution is positive for all values between -∞ and +∞.

You are considering the risk-return profile of two mutual funds for investment.The relatively risky fund promises an expected return of 9%,with a standard deviation of 12%.The relatively less risky fund promises an expected return and standard deviation of 5% and 8%,respectively. A)Which mutual fund will you pick if your objective is to minimize the probability of earning a negative return? B)Which mutual fund will you pick if your objective is to maximize the probability of earning a return between 8% and 12%?

Exhibit 6-1.You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough.The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. Refer to Exhibit 6-1.One sleeping bag you are considering advertises that it is good for temperatures down to 25°F.What is the probability that this bag will be warm enough on a randomly selected May night at the park?

Exhibit 6-6.Let the lifetime of a new Jet Ski be represented by a lognormal variable, where X is normally distributed.Let the mean of the lifetime of the Jet Ski be six years with a standard deviation of three years. Refer to Exhibit 6-6.What proportion of the Jet Skis will last nine or more years?

Examples of random variables that closely follow a normal distribution include the age and the class year designation of a college student.

Exhibit 6-2.Gold miners in Alaska have found,on average,12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces.Assume the amount of gold found per 1000 tons of dirt is normally distributed. Refer to Exhibit 6-2.If the miners excavated 1000 tons of dirt,how little gold must they have found such that they find that amount or less only 15% of the time?

The average wait time at a McDonald's drive-through window is about three minutes ("The Doctor Will See You Eventually," The Wall Street Journal,October 18,2010).Suppose the wait time is exponentially distributed.What is the probability that a randomly selected customer will have to wait no more than five minutes?

The probability density function of a continuous random variable can be regarded as a counterpart of the probability mass function of a discrete random variable.

Let X be normally distributed with mean µ = 25 and standard deviation σ = 5.Find the value x such that P(X ≥ x)= 0.1736.

Let the household income of residents of the United States be represented by Y = e^X,where X is normally distributed.In 2006,the mean U.S.household income was approximately $50,000.Suppose the standard deviation was 16,000.Estimate the proportion of U.S.households that have an income less than 80,000.