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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 7983.
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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View Full Document(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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 Spring '11
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