lec0120 - IEOR 4106: Introduction to OR: Stochastic Models...

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IEOR 4106: Introduction to OR: Stochastic Models Spring 2011, Professor Whitt Class Lecture Notes: Thursday, January 20. The Central Limit Theorem and Stock Prices 1. The Central Limit Theorem (CLT) See Section 2.7 of Ross. (a) Time on My Hands: Suppose that I have a lot of time on my hands, e.g., because I am on a subway travelling the full length of the subway system. Fortunately, I have a coin in my pocket. And now I decide that this is an ideal time to see if heads will come up half the time in a large number of coin tosses. Specifically, I decide to see what happens if I toss a coin many times. Indeed, I toss my coin 1 , 000 , 000 times. Below are various possible outcomes , i.e., various possible numbers of heads that I might report having observed: 1. 500,000 2. 500,312 3. 501,013 4. 511,062 5. 598,372 What do you think of these reported outcomes? How believable are each of these possible outcomes? How likely are these outcomes? We rule out outcome 5; there are clearly too many heads. We rule out outcome 1; it is “too perfect.” Even though 500 , 000 is the most likely single outcome, it itself is extremely unlikely. But how do we think about the remaining three? The other possibilities require more thinking. We can answer the question by doing a normal approximation ; see Section 2.7 of Ross, especially pages 79-83. We introduce a probability model. We assume that successive coin tosses are independent and identically distributed (commonly denoted by IID) with probability of 1 / 2 of coming out heads. Let S n denote the number of heads in n coin tosses. The random variable S n is approximately normally distributed with mean np = 500 , 000 and variance np (1 - p ) = 250 , 000. Thus S n has standard deviation SD ( S n ) = p V ar ( S n ) = 500. Case 2 looks likely because it is less than 1 standard deviation from the mean; case 3 is not too likely, but not extremely unlikely, because it is just over 2 standard deviations from the mean. On the other hand, Case 4 is extremely unlikely, because it is over 20 standard deviations from the mean. See the Table on page 81 of the text.
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(b) The Power of the CLT The normal approximation for the binomial distribution with parameters ( n,p ) when n is not too small and the normal approximation for the Poisson with mean λ when λ is not too small are both special cases of the central limit theorem ( CLT ). The CLT states that a properly normalized sum of random variables converges in distribution to the normal distribution. Of course there are conditions. We give a formal statement; see Theorem 2.2 on p. 79 of
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lec0120 - IEOR 4106: Introduction to OR: Stochastic Models...

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