IEOR 4106: Introduction to OR: Stochastic Models
Spring 2011, Professor Whitt
Class Lecture Notes: Thursday, January 20.
The Central Limit Theorem and Stock Prices
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
Below are various possible outcomes
, i.e., various possible
numbers of heads that I might report having observed:
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
; 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
denote the number of heads in
coin tosses. The random variable
approximately normally distributed with mean
000 and variance
) = 250
has standard deviation
) = 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.