 # Statistics Econometrics Statistics Econometrics Statistics Econometrics Statistics for

• Slides: 13 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch. 11 The Law of Averages 1. The Law of Averages 2. (Law of Large Numbers) 2. The Stochastic Processes 3. Making a Box Model Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics INDEX STATISTICS 1 The Law of Averages (Law of Large Numbers) 2 The Stochastic Processes 3 Making a Box Model Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 2 1. The Law of Averages STATISTICS The Law of Averages (1) Head 100 is tossed Tail Heads with Tails with probability 50% 100 The probability of actual toss to get head To get the third head after getting head twice ? ? ? Head Tail A coin does not remember bygones. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 3 1. The Law of Averages STATISTICS The Law of Averages (2) # of tosses 10 20 30 40 50 60 70 80 90 100 200 300 400 500 # of heads 4 12 16 20 25 32 36 40 46 49 100 159 205 255 # of heads minus half the # of tosses -1 2 1 0 0 2 1 0 1 -1 0 9 5 5 # of tosses # of heads 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 308 353 397 458 509 1017 1518 2021 2532 3030 3524 4018 4521 5055 A run of heads just # of heads minus half the # of tosses 8 3 -3 8 9 17 18 21 32 30 24 18 21 55 doesn’t make tails more likely next time. With a large number of tosses the size of difference between the number of heads and the expected number doesn’t decrease. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 4 1. The Law of Averages STATISTICS The Law of Averages (3) As the number of tosses increases, absolute size of probability error increases. Probability Error # of trials Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 5 1. The Law of Averages STATISTICS Ralative percentage of probability error The Law of Averages (4) But when the number of tosses goes up, this percentage is goes down : the probability error gets smaller relative to the number of tosses. - The Law of Averages - # of trials Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 6 INDEX STATISTICS 1 The Law of Averages (Law of Large Numbers) 2 The Stochastic Processes 3 Making a Box Model Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 7 2. The Stochastic Processes STATISTICS Stochastic Process and Applying a Box Model • To record the number of getting heads in a coin tossing • To see how much the house should expect to win at Roulette • To see how accurate unemployment rate through a sample survey is likely to be 00100 1 Able to analyze using stochastic process If you use a Box Model, you can comprehend comlicated stochastic processes. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 8 2. The Stochastic Processes STATISTICS The Stochastic Processes and Box Model at Inference : A process to find out something about the population using information from the sample The Stochastic Process A Box Model sampling Population box sample drawing tickets drawn tickets ditribution of tickets Population Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 9 INDEX STATISTICS 1 The Law of Averages (Law of Large Numbers) 2 The Stochastic Processes 3 Making a Box Model Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 10 STATISTICS 3. Making a Box Model Box of tickets box drawing tickets 25 draws 13 4 6 2 5 • Pick one ticket at random. And make a note of the number on it. (with replacement) • Put it back in the box. And make a second draw at random again. • Having drawn twice at random with replacement, you add up the two numbers. 32462 35442 36412 41556 22255 Ten times repeated results sum : 88 88 84 80 90 83 78 95 94 80 89 In principle, the sum could have been as small as 25 or as large as 150. But in fact, the ten observed values are all between 75 and 100. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 11 STATISTICS 3. Making a Box Model 1 2 3 4 5 6 1 What numbers go into the box? 2 13 4 6 2 5 How many of each kind? 3 25 draws How many draws? Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 12 STATISTICS 3. Making a Box Model Example A gambling problem in which the same bet is made several times 1 Which ticket? 2 How many of each kind? The tickets in the box show the amounts that can be won (+) or lost (-) on each play. The chance of drawing any particular value from the box 3 equals the chance of How many draws? winning that amount on a single play. 4 The number of draws equals Net gain? the number of plays. +1000 -1000 +1000 0 +1 1+0100 -1 -0100 000 +1000 + -1000 …… Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 13