Chapter 2: FAQs 31
What is the Central Limit Theorem
and What are its Implications for
Finance?
Short Answer
The distribution of the average of a lot of random num-
bers will be normal (also known as Gaussian) even
when the individual numbers are not normally dis-
tributed.
Example
Play a dice game where you win $10 if you throw a six,
but lose $1 if you throw anything else. The distribution
of your profit after one coin toss is clearly not normal,
it’s bimodal and skewed, but if you play the game thou-
sands of times your total profit will be approximately
normal.
Long Answer
LetX 1 ,X 2 ,...,Xnbe a sequence of random variables
which are independent and identically distributed (i.i.d.),
with finite mean,mand standard deviations.Thesum
Sn=
∑n
i= 1
Xi
has meanmnand standard deviations
√
n. The Central
Limit Theorem says that asngets larger the distribution
ofSntends to the normal distribution. More accurately,
if we work with the scaled quantity
Sn=
Sn−mn
s
√
n
then the distribution ofSnconverges to the normal
distribution with zero mean and unit standard devia-
tion asntends to infinity. The cumulative distribution
