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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch. 11 The Law of Averages 1.The Law of Averages (Law of Large Numbers) 2.The Stochastic Processes 3.Making a Box Model

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/13 INDEX 1 The Law of Averages (Law of Large Numbers) 2 The Stochastic Processes 3 Making a Box Model

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/13 \100 1. The Law of Averages The Law of Averages (1) Head \100 Tail Heads with probability 50% Tails with probability 50% \100 is tossed HeadTail ??? The probability of actual toss to get head \100 To get the third head after getting head twice A coin does not remember bygones.

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/13 With a large number of tosses the size of difference between the number of heads and the expected number doesn ’ t decrease. A run of heads just doesn ’ t make tails more likely next time. The Law of Averages (2) 1. The Law of Averages # of tosses # of heads # of heads minus half the # of tosses # of tosses # of heads # of heads minus half the # of tosses 1046003088 201227003533 30161800397-3 402009004588 5025010005099 603222000101717 703613000151818 804004000202121 904615000253232 100496000303030 20010007000352424 30015998000401818 40020559000452121 500255510000505555

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/13 # of trials ProbabilityError The Law of Averages (3) As the number of tosses increases, absolute size of probability error increases. 1. The Law of Averages

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/13 # of trials Ralative percentage of probability error 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 - The Law of Averages (4) 1. The Law of Averages

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/13 INDEX 1 2 The Stochastic Processes 3 Making a Box Model The Law of Averages (Law of Large Numbers)

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/13 2. The Stochastic Processes 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 Able to analyze using stochastic process 100 \100 100 \100 100 \100 100 \100 If you use a Box Model, you can comprehend comlicated stochastic processes.

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/13 The Stochastic Processes and Box Model at Inference Inference : A process to find out something about the population using information from the sample The Stochastic ProcessA Box Model box ditribution of tickets drawing tickets drawn tickets Population sampling sample 2. The Stochastic Processes

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/13 INDEX 1 The Law of Averages (Law of Large Numbers) 2 The Stochastic Processes 3 Making a Box Model

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/13 3. Making a Box Model Ten times repeated results 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. Pick one ticket at random. And make a note of the number on it. 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. box 2 3 4 6 5 drawing tickets 25 draws (with replacement) 1 32462 35442 36412 41556 22255 sum : 88 Box of tickets

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/13 Making a Box Model 2 3 4 6 5 1 25 draws 123456 What numbers go into the box? 1 How many of each kind? 2 How many draws? 3 3. Making a Box Model

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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/13 Which ticket? 1 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 equals the chance of winning that amount on a single play. The number of draws equals the number of plays. Example How many of each kind? 2 How many draws? 3 Net gain? 4 +1000-1000 A gambling problem in which the same bet is made several times -1000 +1000 -1000 + …… 3. Making a Box Model

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