Exam 6: Continuous Probability Distributions

Let house prices in a rich community in Chicago be represented by Y= e^x where X is normally distributed. Suppose the mean house price is $1.8 million and the standard deviation is $0.4 million. What is the proportion of the houses that are worth more than $2.5 million?

On a particular busy section of the Garden State Parkway in New Jersey, police use radar guns to detect speeding drivers. Assume the time that elapses between successive speeders is exponentially distributed with the mean of 15 minutes. What is the probability of a waiting time less than 10 minutes between successive speeders?

The normal distribution is ________ in the sense that the tails get closer and closer to the horizontal axis but never touch it.

For any normally distributed random variable with mean μ and standard deviation σ, the percent of the observations that fall between [μ - 2σ, μ + 2σ] is the closest to ________.

Find the following probabilities for a standard normal random variable Z. A) P(-0.25 < Z ≤ 0.50) B) P(-2.5 ≤ Z < 0) C) P(-1.45 < Z < 3.09) D) P(Z < -1.75)

A normal random variable X has a mean of 17 and a variance of 5. A) Find the value x for which P(X ≤ x) = 0.0020. B) Find the value of x for which P(X > x) = 0.0122.

The annual return of a well-known mutual fund has historically had a mean of about 10% and a standard deviation of 21%. Suppose the return for the following year follows a normal distribution, with the historical mean and standard deviation. What is the probability that you will lose money in the next year by investing in this fund?

The stock price of a particular asset has a mean and standard deviation of $58.50 and $8.25, respectively. Use the normal distribution to compute the 95th percentile of this stock price.

On average, a certain kind of kitchen appliance requires repairs once every four years. Assume that the times between repairs are exponentially distributed. What is the probability that the appliance will work at least six years without requiring repairs?

What does it mean when we say that the tails of the normal curve are asymptotic to the x axis?

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.

An investment consultant tells her client that the probability of making a positive return with her suggested portfolio is 0.90. What is the risk, measured by standard deviation that this investment manager has assumed in her calculation if it is known that returns from her suggested portfolio are normally distributed with a mean of 6%?

Find the following probabilities for a standard normal random variable Z. A) P(0 < Z ≤ 1.96) B) P(-1.96 ≤ Z < 0) C) P(-1.96 < Z ≤ 1.96) D) P(Z ≥ 1.96)

The time for a professor to grade a homework in statistics is normally distributed with a mean of 12.6 minutes and a standard deviation of 2.5 minutes. What is the probability that randomly selected homework will require more than 12 minutes to grade?

The average annual percentage rate (APR) for credit cards held by U.S. consumers is approximately 15 percent ("Ouch - Credit Card APR Now Tops 15 Percent," Time, January 3, 2012). Suppose the APR for new credit card offers is normally distributed with a mean of 15% and a standard deviation of 4%. What APR must a credit card charge to be in the bottom 10% of all cards?

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. Find the mean and standard deviation of the waiting time. A) 115 seconds and 49.07 seconds B) 1.15 minutes and 0.4907 minutes C) 1.15 minutes and 24.08333 (minute)^{2 D)} 115 seconds and 2408.3333 (second)²

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?

Cumulative distribution functions can only be used to compute probabilities for continuous random variables.

If Y = e^X has a lognormal distribution, what can be said of the distribution of the random variable X?

Suppose the amount of time customers must wait to check bags at the ticketing counter in Boston Logan Airport is exponentially distributed with a mean of 14 minutes. What is the probability that a randomly selected customer will have to wait more than 20 minutes?