Exam 5: Continuous Random Variables and Probability Distributions

If X is a normal random variable with mean μ = 4 and standard deviation σ = 2,and Y is a normal random variable with mean μ = 10 and standard deviation σ = 5,then the probabilities P(X < 0)and P(Y < 0)are equal.

The cumulative distribution function for a random variable X,F(x₀),is the area under the probability density function f(x)up to x₀.

If the random variable X is exponentially distributed with parameter λ = 4,then the probability P(X ≤ 0.25)is equal to:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let the random variable X follow a normal distribution with a mean of 17.1 and a standard deviation of 3.2. -What is P(15 < X < 20)?

The number of viewers ordering a particular pay-per-view program is normally distributed.20% of the time,fewer than 20,000 people order the program.Only ten percent of the time more than 28,000 people order the program.What is the mean and standard deviation of the number of people ordering the program?

The mean and median are the same for a random variable that is exponentially distributed.

If X is a normal random variable with mean of 2 and standard deviation of 5,then P(X < 3)= P(X > 7).

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Suppose that the time between successive occurrences of an event follows an exponential distribution with mean 1/λ minutes.Assume that an event occurs. -Show that the probability that more than 8 minutes elapses before the occurrence of the next event is e^-8λ.

Let the random variable Z follow a standard normal distribution,and let z₁ be a possible value of Z that is unknown but identifiable by position and area.Find z₁ if the symmetrical area between a negative z₁ and a positive z₁ is 0.9544.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: The length of time it takes to be seated at a local restaurant on Friday night is normally distributed with a mean of 15 minutes and a standard deviation of 4.75 minutes. -What is the probability that you have to wait more than 20 minutes to be seated?

The shape of the uniform probability distribution is:

A homeowner has installed a new energy-efficient furnace.It is estimated that over a year the new furnace will reduce energy costs by an amount that can be regarded as a random variable with a mean of $265 and standard deviation of $55.Stating any assumptions you need to make,find the mean and standard deviation of the total energy costs reductions over a period of 5 years.

Let the random variable Z follow a standard normal distribution,and let z₁be a possible value of Z that is representing the 75th percentile of the standard normal distribution.Find the value of z_1.

If the continuous random variable X is uniformly distributed over the interval 6 to 10,then the height of this distribution,f(x),is:

In the normal distribution,the total area beneath the curve represents the probability for all possible outcomes for a given event.

The mean of a probability distribution provides a measure of the center of the distribution.

Given that Z is a standard normal random variable,P(-1.2 ≤ Z ≤ 1.5)is:

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: The amount of time you have to wait at a dentist's office before you are called in is uniformly distributed between zero and twenty minutes. -What is the probability that you have to wait more than 8 minutes?

The shape of the normal probability distribution is determined by the population standard deviation σ.Small values of σ reduce the height of the curve and increase the spread;large values of σ increase the height of the curve and reduce the spread.

The proportion of the total area under the normal curve that lies within ± one standard deviation of the mean is approximately 0.75.