dents, they complain, have to commute too far to scho ity officials respond with the following information: the mean dist mmuted is 3.2 miles. suming that the university officials' inform ents about the distribution of commute distances for students at this university. (a) According to Chebyshev's theorem, at least (Choose one) V of the (Choose one) commute distances lie between 10.6 miles and 23.4 miles. (b) According to Chebyshev's theorem, at least - (about 89%) of the commute distances lie between miles and miles. (Round your answer to 1 decimal place.) Fill in the P (X = x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are -6, 1, 3, 5, and 6. Value x of X P(X = x) 0.24 0.28 0.17 3. Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" (g) and "boys" (b), which we write gog, bob, etc. For each outco unting the number of girls in each outcome. For example, if the outcome is gbg, then R(gbg) = 2. Suppose th hed in terms of R as follows: X= 2R-R' -4. he values of X" are given in the table below Outcome ggg bgb bbb bbg ggb gog gbb bag Value of X -7 -3 -4 -3 - 4 - 4 - 3 - 4 Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value x of X 0 0 0 Py (x) 0 0 0 Let X be a random variable with the following probability distribution. Value x of X P(X=x) (a) Find the expectation E (X') of X -10 0.60 E (X) = 1 (b) Find the variance Var(X') of X. o 0.05 var (x) = 0 10 0.05 20 0.30 . Suppose that we've decided to test Cla ychic Center, to see if she really has psychic abilities. while talking to her made up of 13 hearts, 13 spades, 13 diamonds, and 13 clubs) and draw suit (heart, spade, diamond, or club) of the card we drew. After getting her guess, we'll return the card to th card, and get her guess for the suit of this second card. We'll repeat this proce esses for each. Assume that Clara is not clairvoyant, that is omly guesses on each card. Answer the following. imate the number of cards in the sample for which Clara correctly guesses the suit build your suit by giving the mean of the ant distribution (that is, the expectation of b) Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response 5. The workers' union at a part university belong to the workers' union. Recently, the workers went on strike, and n orkers (chosen at random) at the university to get their opinions on the strike. What is the probability that exactly 3 of the workers interviewed are union members? Round your response to at least three decimal places . From experience, an airline knows that only 70% of the passengers booked for a certain flight actually show up. If 9 passengers are randomly selected, find the probability that more than 7 of them show up. Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places. 0 B. There is some evidence that, in the years 1981 - 85, a simple name change resulted in a short-term increase in the price of certain business firms' stocks (relative to the pri forsky and P. Swyngedouw, "Does it pay to change your company's name? A stock market persp ce v.6, pp. 320 - 35, 1987.) Suppose that, to test the profitability of name changes in the more recent market (the past five years), we analyze the stock prices of a large sample of corporations shortly after they changed names, and we find that the me crease in stock price was about 0.73%, with a standard deviation of 0.16%. Suppose that this mean and standard deviation apply to the population of all companies that changed names during the past five years. Complete the following state that changed names during the past five years. a) According to Chebyshev's theorem, at least 84% of the relative increases in stock price lie between |% and %. (Round your answer to 2 decimal places.) (Choose one) V (b) According to Chebyshev's theorem, at least (Choose one) V of the relative increases in stock price lie between 0.41% and 1.05% 89% 9. Fill in the P (X= x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are 1, 2, 3, 4, and 5. Value x of X P(X = x) 0.10 0 0.15 0 0.18 10. A machine that manufactures auto estimated to produce a defective piston 1% of the time. Suppose that this estimate is correct and that a andom sample of 90 pistons produced by this machine is taken. Answer the following (a) Estimate the number of pistons in the sample that are defective by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response. 0 (b) Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places