Exam 7: Random Variables and Probability Distributions

If random variablesx1and x2 are independent, thenσ22x12+xx=+σσ1 2x2

A random variable is continuous if the set of possible values includes an entire interval on the number line.

In a study of the reaction times of left- and right-handers' reaction times to tones delivered to the right ear, the right-handers' scores were approximately normally distributed with a mean of 240 milliseconds and standard deviation of 32 milliseconds.The mean score for right-handers was 210 ms. a)Sketch a normal distribution that describes left-handers' reaction times, and locate the mean reaction time for right-handers in this distribution. b)About what proportion of left-handers reaction times would be "better" (i.e.smaller) than the mean reaction time for right-handers?

Ifx is a random variable, and random variableyis defined as follows,

For a continuous random variable x, the height of the density curve over an interval a to b represents the probability that x is between a and b.

In a certain very large city, the Department of Transportation (D.O.T.)has organized a complex system of bus transportation.In an advertising campaign, citizens are encouraged to use the new "GO-D.O.T!" system and head for the nearest bus stop to be transported to and from the central city.Suppose that at one of the bus stops the amount of time (in minutes)that a commuter must wait for a bus is a uniformly distributed random variable, T.The values of T run from 0 minutes to 30 minutes. a)Sketch the probability distribution of T. b)What is the probability that a random commuter will spend more than 12 minutes waiting for GO-D.O.T?

For a variable z that has a standard normal distribution, a)W c)What is the probability that z is between -+1.43 and 1.43 d)What value of z separates the smaller 10% of the standard normal distribution from the larger 90%? e)

While playing Monopoly, Laura estimated the probabilities of the non-zero rents according to the following probability distribution: Consider the random variable x = dollar amount in rent in a Monopoly roll. a)If the table above specifies the probability distribution f ()x , what is the mean of the random variable x? b)If the probabilities are associated with the outcomes as in the table above, what is the standard deviation of the random variable x?

Inspecting meat is a very important part of the process of ensuring healthy food.So that meat inspectors do not fall into a pattern of inspection, they use a 4-sided die to decide whether or not to sample the meat from a particular "side" of beef.The meat inspector rolls the die and if it comes up a 4, a sample is taken of that meat.Forty sides of beef are available for possible inspection.Define random variable x = number of sides selected for inspection out of the 40 available. a)What is the mean of the random variable x? b)What is the standard deviation of the random variable x?

In a major study by the statistics classes at Jefferson High School, parking spaces were examined for compliance with the requirement to put money in the meters.Overall, the students found that 80% of metered parking places had meters that had not expired, and 20% had expired.If the traffic officer in charge of ticketing cars with expired meters, and starts at a random location in the City, what is the probability he or she will find an expired meter before the 3rd one?

For every random variable,

For a discrete random variable

A random variable is discrete if the value of the random variable depends upon the outcome of a chance experiment.

The distribution of all values of a random variable is called a normal distribution.

A normal probability plot suggests that a normal probability model is plausible if there is no obvious pattern in the scatter of points.

One common method used to forecast the results of elections is to take an "exit poll," asking people who they supported as they come out of the voting place.Suppose Smith and Jones are running for City Dogcatcher.In point of fact, 60% of the voting public in the neighborhood of the polling place support Smith.Assuming that the candidate of choice for a voter leaving the polling place is independent of the candidate of choice for the next person to leave, what is the probability that the first 5 voters to leave the polling place would have this sequence for candidate of choice? Smith, Smith, Smith, Jones, Smith

For a discrete random variable,

In the old Roman Coliseum, two horses would be placed in tandem (side by side)and hitched to a chariot.Since the fast chariots needed to be able to pass the slow chariots, it was of some importance that the horses have room to run, but also they should not be too large.For chariot hitching purposes, the widest measure across a horse occurs in the rump area.The mean rump width is approximately 27" with a standard deviation of about 2".Let random variable w = width in inches across the rump of a randomly selected horse a)The usual length measure as it applies to horses is the "hand," which is by definition equal to 4".Define random variable h = width in hands across the rump of a randomly selected horse.What are the mean and standard deviation of h?