Chapter 5: Probability and Random Variables

Major Hurricanes. The random variable Y is the number of person living in a randomly selected occupied housing unit. Its probability distribution is as follows.

Part (a) Find and interpret the mean of the random variable.

Part (b) Obtain the standard deviation of the random variable by using one of the formulas given in definition 5.10

Part (c) Construct a probability histogram for the random variable, locate the mean: and show one, two, and three standard deviation intervals.

Children's Gender. The random variable X is the number of girls of four children born to a couple that is equally likely to have either a boy or a girl. Its probability distribution is as follows.

Part (a) Find and interpret the mean of the random variable.

Part (b) Obtain the standard deviation of the random variable by using one of the formulas given in definition 5.10

Part (c) Construct a probability histogram for the random variable, locate the mean: and show one, two, and three standard deviation intervals.

Dice. The random variable Y is the sum of the dice when two balanced dice are rolled. Its probability distribution is as follows

Part (a) Find and interpret the mean of the random variable.

Part (b) Obtain the standard deviation of the random variable by using one of the formulas given in definition 5.10

Part (c) Construct a probability histogram for the random variable, locate the mean: and show one, two, and three standard deviation intervals.

World series. The World series in baseball is won by the first team to win four games ( ignoring the 1903 and 19919 - 1921 World series, when it was a best of nine). From the document World series history on the baseball Almanac website. as of November 2013, the length of the world series are as given in the following table.

Let X denote the number of games that it take to complete a world series, and let Y denote the number of games that it took to complete a randomly selected world series from among those considered in the table.

Part (a) Determine the mean and standard deviation of the random variable Y. Interpret your resuts.

Part (b) Provide an estimate for the mean and standard deviation of the random variable X. Explain your reasoning

Archery. An archer shoots an arrow into a square target 6 feet on a side whose center we call the origin . The outcome of this random experiment is the point in the target hit by the arrow. The archer score 10 points if she hits the bull's eye - a disk of radius 1 foot centered at the origin, she score 5 points otherwise. Assume that the archer will actually hit the target and is equally likely to hit any portion of the target . For one arrow shot, Let S be the score. A probability distribution for the random variable S is as follows.

Part (a) On average, how many points will the archer score per arrow shots?

Part (b) Obtain and interpret the standard deviation of the score per arrow shot.

Playing Cards. An ordinary deck of playing cards has 52 cards. There are four suits-spades, hearts, diamonds, and clubs- with 13 cards in each suit. Spades and clubs are black; hearts and diamonds are red. If one of these cards is selected at random, what is the probability that it is

(a). a spade? (b). red? (c). not a club?

All number Passwords. The technology consultancy Data genetics published the online document PIN analysis. In addition to analyzing PIN numbers, passwords trends were examined. seven million number passwords were collected and yielded the following estimate of the probability distribution of the number of digit used in an all numeric password.

On average, approximately how many digit would expect an all numeric password to have ? Explain your answer.

Roulette. An American roulette wheel contains 38 numbers: 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally likely to land on any of the 38 number. You win \(1: otherwise you lose your \)1. Let X be the amount you win on your \(1 bet. Then x is a random variable whose probability distribution is as follows.

Part (a) Verify that the probability distribution is correct.

Part (b) Find the expected value of the random variable X.

Part (c) On average, how much will lose per play?

Part (d) Approximately how much would you expect to lose if you bet \)1 on red 100 times? 1000 times?

Part (e) Is roulette a profitable game to pay ?Explain

Evaluating Investments. An investor plans to put $50.000 in one of four investments. The return on each investment depends on whether next year's economy is strong or weak. The following table summarizes the possible payoffs. in dollars for the four investments.

Let V, W, X and Y denotes the payoffs for the certificate or deposit office complex, land speculation. and technical school, respectively the V, W, X and Y are random variables . assume that nest year's economy has a 40% chance of being strong and a 60% chance of being weak.

Part(a) Find the probability distribution of each random variable V, W, X, and Y

Part (b) Determine the expected value of each random variable.

Part (c) Which investment has the best expected payoffs? the worst?

Part (d) Which investment would you select? Explain

Homeowner's Policy . An insurance company wants to design a homeowner's policy for mid priced homes. From data compiled by the company. it is known that the annual claim amount, X in thousands of dollars, per homeowner is a random variable with the following probability distribution.

Part (a) Determine the expected annual claim amount per homeowner.

Part (b) How much should the insurance company charge for the annual premium if it wants to average a net profit of $50 per policy?