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# Statistics Measurement in Economics ECON 310

UW

GPA 3.6

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This 4 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 310 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/205161/econ-310-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.

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Date Created: 09/17/15

1 to 03 q Section 5 ECON 310 Spring 2007 March 8 2007 Question 5 suggested exercises Lecture notes 10 Let X be a continuous random variable that represent the length of time until the next hit on a website in 100 minutes units Suppose that X is exponentially distributed with PX gt 10 08 Find the mean and standard deviation of X Question 5 Chapter 8 Sandholm and Saraniti Large electronics retailers earn enormous pro ts from selling extended warranties on the products they carry At one seller plasma screen televisions come with a one year manufacturer s warranty and the retailer offers an additional three year ex tended warranty that begins after the rst year Assume the rate at which a plasma screen television breaks down can be described by a random variable T N exponential125 a Determine the probability that a plasma screen television breaks down during the rst year b Determine the probability that a new plasma screen television will break down during the extended warranty period ie during the second third or fourth year c Determine the probability that a plasma screen television that has lasted one year will break down during the extended warranty period Your mail order rm employs a large number of operators to take phone orders When Alvin begins a phone order the amount it time it takes for him to complete the order follows an exponential distribution with rate 25 so that his expected time to complete an order is 4 minutes Similarly Bertha completes orders at an exponential rate of 2 and Cedric completes orders at an exponential rate of 18 The amounts of time it takes to complete orders are independent across operators a Suppose that Alvin begins a phone order and that after 5 minutes the order is not yet complete Conditional on this event what is the probability that he will not complete the order until 10 or more minutes have passed in total b What is the probability that Alvin completes his next order in 2 minutes or less What about Bertha Cedric Suppose all three operators begin taking an order simultaneously c What is the probability that the rst operator completes his or her order in 2 minutes or less d What is the probability that Cedric completes his order rst e What is the probability that Bertha completes her order rst doing so in 15 minutes or less Hint Use the fact that the time required to complete the rst order and the identity of the operator who completes it are independent of one another Question 1a d suggested exercises Lecture notes 11 True or false Justify a If Z N N0 1 then i EZ2 1 P7oo Z oo1 iii P717 Z 17 2P0 Z 17 iv P71 Z 0z b leN01andYN01thenWXYN01aswell c If Y N Nu 02 then PY 2 0 PY 0 5 5 a T 00 d Suppose that X N N10 20 and Y 10 7 2X then Y N N710 780 Sketch the standard normal density function indicating the values 3 2 1 0 1 2 3 on the horizontal axis Then determine the following probabilities a P0 Z 1 b P0 Z 15 c P0 Z 2 d P0 Z 25 e P198 Z 49 f P52 Z 122 g P175 Z 7104 Suppose that Z is a standard normal random variable Compute the value of the realization 2 such that a The probability that Z is less than 2 is 2119 b The probability that Z lies between 72 and z is 9030 c The probability that Z lies between 72 and z is 2052 d The probability that Z is less than 2 is 9948 e The probability that Z is greater than 2 is 6915 An auto parts dealer describes his companys dollar sales on weekdays Saturdays and Sundays using the random variables X1 X2 and X3 where X1 N N12 000 4 000 000 X2 N N20000 10 000 000 X3 N N11000 8 000 000 a Compute the probability that sales are less than 10000 on the coming Monday on the coming Saturday and on the coming Sunday b Compute the probability that sales are more than 18000 on the coming Monday on the coming Saturday and on the coming Sunday c Average sales on weekdays are higher than average sales on Sundays Does this imply that the probability of a very high sales gure on a weekday exceeds the probability of a very high sales gure on a Sunday Explain A steel company manufactures steel in two plants one in Bethlehem Pennsylvania the other in Gary Indiana The daily output of the Pittsburgh plant is described by the random variable B N N55 30 while the daily output of the Cary plant is described by the random variable G N N38 25 Units are tons in each case Finally production levels at the two plants are independent of one another a What is the probability that on Tuesday the Bethlehem plant produces at least 60 tons of steel and the Cary plant produces at least 40 tons of steel b What is the probability that on Wednesday the two plants produce more than 100 tons of steel in total c What is the probability that on Thursday the Bethlehem plant produces at least 20 more tons of steel than the Cary plant to 03 Section 7 ECON 310 Spring 2007 March 287 2007 Law of Large Numbers If you want to gamble at a game of pure chance7 you are better off playing craps than roulette The probability of winning a bet on red or black in roulette is z 4 resulting in an expected return on a one dollar bet of 70526 If you place a one dollar bet on Don t Pass77 at the craps table7 you win a dollar with probability 4930 otherwise you lose your dollar a You plan to bet one dollar at a time at craps Let Xi represent the returns on your 2 bet What is the distribution of X1 What is its expected value and variance b What are the random variables which represent your total winnings after n bets and your winnings per bet after n bets What are the expected value and variance of these random variables c What does the Weak Law of Large Numbers say about your total winnings after a large number of bets d What does the Weak Law of Large Numbers say about your winnings per bet after a large number of bets How large is Large e Suppose you place 100 bets What is the probability that your total winnings are positive What is the probability that your winnings per bet are positive f Suppose you make 100 trips to the casino7 placing 100 bets each time What is the probability that your total winnings are positive What ranges of total winnings represent the best 507 107 57 and 1 of your possible outcomes g In Las Vegas casinos7 craps is a far more popular casino game than roulette Why do you think this is so Distribution of the Sample Mean Let X be a random variable with the following distribution a What is EX b Suppose you draw a random sample of 2 observations from the above distribution What are the different possible values of sample mean c What is the distribution of the sample mean d Now suppose we have a random variable which takes 1000 values instead of 3 Will it be possible to derive the distribution of the sample mean using the approach in part 2c ls is easy e If we don t know the distribution of X7 will it be possible to derive the distribution of the sample mean using the approach in part 2c True or false Justify a If Xi1 is a sequence of iid random variables with u and VXl 02 then i Y is distributed approximately NW 0271 F 01 ii my 7 p i N0a2 iii Sn X1 Xn is distributed approximately standard normal b Let Xi1 be a sequence of iid random variables with mean EXt 0 and VXt 1 Then in is approximately 0 for T sufficiently large c If X is a random variable with mean u and variance 02 then the percentage of realizations that lie within k standard deviations of the mean ie within pi k0 must be at least 10017 1k2 Central Limit Theorem You have a coin whose bias is 3 which you plan to toss 150 times Answer the following questions about tosses of this coin using the normal approximation to the binomial a What is the probability that between 40 and 60 tosses come up heads b What is the probability that 30 or fewer tosses come up heads Poisson Limit Theorem A manufacturer of hazardous chemicals is acutely concerned with acci dents in his factory During each month7 there are 9000 independent opportunities for accidents to occur7 and that each opportunity leads to an actual accident with probability 00007 a Compute the exact probability that 07 17 27 or 3 accidents occur next month b Approximate these same probabilities using the Poisson limit theorem c Use the Poisson limit theorem to estimate the probability that 9 or more accidents occur during the next year

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