Exam 6: Continuous Probability Distributions

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. 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?

A continuous random variable is characterized by uncountable values and can take on any value within an interval.

Let X be normally distributed with mean µ = 250 and standard deviation σ = 80. Find the value x such that P(X ≤ x) = 0.9394.

Jennifer is waiting for a taxicab. The average wait time for a taxi is six minutes. Suppose the wait time is exponentially distributed. What is the probability that a taxi arrives in three minutes or less?

The mean and standard deviation of the continuous uniform distribution are equal.

Compute the mean and variance of a lognormal variable Y if the mean and the variance of the underlying normal variable are μ = 2; σ² = 1.8.

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

Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon. What is the probability a randomly chosen gas station charges more than $3.70 per gallon?

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. What is the probability a randomly selected patient will have to wait more than 10 minutes?

We are often interested in finding the probability that a continuous random variable assumes a particular value.

It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the probability P(X < 20) related to P(X < 16)?

The exponential distribution is related to the Poisson distribution.

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take more than five hours to construct a soapbox derby car.

Gold miners in Alaska have found, on average, 12 ounces of gold per 1,000 tons of dirt excavated with a standard deviation of 3 ounces. Assume the amount of gold found per 1,000 tons of dirt is normally distributed. What is the probability the miners find more than 16 ounces of gold in the next 1,000 tons of dirt excavated?

We can use the ________ transformation, x = µ + zσ, to compute x values for given probabilities.

Given that the probability distribution is normal, it is completely described by its mean μ > 0 and its standard deviation σ > 0.

It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the probability P(X > 24) related to P(X < 16)?

The average annual inflation rate in the United States over the past 98 years is 3.37% and has a standard deviation of approximately 5% (Inflationdata.com). In 1980, the inflation rate was above 13%. If the annual inflation rate is normally distributed, what is the probability that inflation will be above 13% next year?

Just as in the case of the continuous uniform distribution, the probability density function of the normal distribution may be easily used to compute probabilities.

The average time between trades for a high-frequency trading investment firm is 40 seconds. Assume the time between trades is exponentially distributed. What is the probability that the time between trades for a randomly selected trade and the one proceeding it is more than a minute?