Statistics Measurement in Economics
Statistics Measurement in Economics ECON 310
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This 6 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 Ignacio Monzon in Fall. Since its upload, it has received 17 views. For similar materials see /class/205135/econ-310-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.
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Date Created: 09/17/15
ECON 310 TA SECTION WEEK 9 prepared by SANG YOON TIM LEE and lGNACIO MONZON I LLN and CLT For a sequence of iid random variables Xi1 with mean i4 and variance 72 the sum 5 and sample mean Xn are and ESn quotVI 0311 i4 WINS n02 Vm39Xn The sample mean is crucial for LLN and CLT Law of Large Numbers P 59 E ii 7 64 e gt 1 as n gt oo Coin ips ip twice don t necessarily get exactly one heads and one tails But when you ip it more and more times you do expect that you ll get heads about one half of the time Unfortu nately LLN does not tell you exactly how many times you have to do this for that to happen LLN tells us that the sample mean converges to some number On the other hand CLT tells us that the sample mean converges to some distribution De nition For a sequence of random variables 1981 and a continuous random variable L the sequence converges to L in distribution and we write Y 4 Lquot if for any interval a b PY E ub gt PL E Hb asn gt oo Coin ips again ip the coin 10 million times count the fraction of heads we get do that again and again 10 million times Histogram will look much like a normal distribution centered at Central Limit Theorem For a sequence of iid random variables Xil1 with mean E Xi y and variance VarX 72 514 ltX Fgt adZ asnaoo where Z N N01 For 71 large enough 5 m Nn4m72 and X m N4 As with LLN CLT does not tell us how large 71 should be for this to be true VS Chapter 9 Exercise 6 An IRS computer ags suspicious tax returns which are then looked over by an IRS agent to determine whether an audit is needed The probability that a agged return requires an audit is 40 Suppose that Agent Anderson evaluates 100 agged returns per week 92 What is the approximate probability distribution for 5100 the number of agged returns that require an audit this week F Determine the probability that this week Agent Anderson finds between 35 and 40 returns inclusive that require an audit 0 Determine the probability that Agent Anderson finds more than 50 returns that require an audit VS Chapter 9 Exercise 10 Two sales reps Amy and Beatrice work for a large shoe and apparel manufacturer Their company is trying to convince retailers to allocate space to a new product line The sales rep that secures the most new contracts with retailers this month earns a 10000 bonus Amy will convince a retailer to accept the new product line with a probability of 25 while Beatrice is successful with probability 30 Each rep s sales calls can be modeled a sequence of iid random variables and both reps have 100 retailers in their sales territory 9 Describe the approximate probability distributions of S A and S B the total numbers of suc cessful sales calls by Amy and Beatrice U Determine the probabilities that each rep has between 25 and 30 successful sales calls F What does the random variable S A 7 53 represent What is its approximate distribution 53 What is the probability that Amy wins the bonus II Exponential Distribution Normal distribution Bell curve exponential distribution waiting times Time you wait for a bus the next customer someone else to drop the course so you can enroll 111 De nition The distribution function of a exponential random variable T is completely characterized by one parameter only A ex iAt if t 2 0 f t p 0 otherw1se A describes how long you are likely to wait A large means shorter waiting time 112 Traits If T N expA 1 PT g t 1 iexp7At PT gt t exp7At 2 ET me 71 3 KY CT then Y exp g 4 Memorylessness PT gt s t T gt s PT gt t Memorylessness might not come immediately For example you arrived at a bus stop and think a bus might come in about 2 minutes After a minute you still think the bus might come in about 2 minutes 113 VS Chapter 8 Exercise 6 Your mail order firm employs a large number of operators to take phone orders When Alvin begins a phone order the amount of time it takes for him to complete the order follows an ex ponential 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 com plete 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 F What is the probability that the first operator completes his or her order in 2 minutes or less 53 What is the probability that the first ashlight to die dies within six months CD What is the probability that Bertha completes her order first doing so in 15 minutes or less Hint Use the fact that the time required to complete the first order and the identity of the operator who completes it are independent of one another 114 VS Chapter 8 Exercise 7 John owns two ashlights one amber and one blue The lifetime of each ashlight has an expo nential distribution The expected lifetime of the amber ashlight is 15 years while the expected lifetime of the blue ashlight is 25 years 9 What is the probability of this combination of events the amber ashlight lasts for less than 3 years and the blue ashlight lasts for more than 3 years F Suppose that the blue ashlight lasts for 10 or more years What is the probability that it lasts between 9 and 12 years in total 0 What is the probability that the amber ashlight lasts longer than the blue ashlight 53 What is the probability that the first ashlight to die dies within six months ECON 310 TA SECTION WEEK 7 prepared by SANG YOON TIM LEE and lGNACIO MONZON Uniform Distribution Let X be distributed uniformly that is X N Ulh Then 0 Density function f X i hl 2 0 Expected Value E X o Variance VmX Problems 1 Half wit Enterprises is a firm that produces certain toys and has a demand represented by the function X 7 where P is the unit price and I stands for the consumers income which is uniformly distributed between 1000 dollars and 2 000 dollars per week a Find the firms expected sales per week b If total costs TC are fixed and equal to 1 000 dollars find the expected profits of this firm c Find the standard deviation of the firm s profits 2 Let X N U7kk where k is a positive parameter Find k when a W gt 1 b M 07 9 Let F be a random variable denoting tomorrow s temperature in Fahrenheit The weather man announces that F N U3040 a What is the probability that tomorrow s temperature is over 38 Fahrenheit b What is the probability that tomorrow s temperature is between 32 and 36 Fahrenheit c What is the distribution of C tomorrow s temperature in Celsius What is the proba bility that tomorrow s temperature is below 0 Celsius F C 32 4 In a second price auction the object is awarded to the highest bidder but the payment is only equal to the second highest bid It is a known fact in game theory that bidders bid exactly how much they actually value the object Suppose the Madison Museum of Metropolitan Art is holding a second price auction for an antique selection There are N people participating in the auction and the only fact that they know about each other is that everyone values the selection at a dollar value between 0 and 1 In other words each person thinks that everyone else s value of the selection 390 is a uniform random variable in the interval 01 and so 390 U01 a Suppose N 2 What is the cumulative probability density function expected value and variance of the winning bid b For a general N what is the cumulative probability density function expected value and variance of the winning bid