Anon

346 The Basics of financial economeTrics

Property 2. The distribution is stable under summation. That is, if X has

a normal distribution F, and X 1 ,... ,Xn are n independent random

variables with distribution F, then X 1 +... + Xn is again a normal

distributed random variable.

In fact, if a random variable X has a distribution satisfying Properties 1 and
2 and X has a finite variance, then X has a normal distribution.
Property 1, the location-scale invariance property, guarantees that
we may multiply X by b and add a where a and b are any real numbers.

Then, the resulting a + b (^) ⋅ X is, again, normally distributed, more pre-
cisely, N (a + μ, bσ). Consequently, a normal random variable will still be
normally distributed if we change the units of measurement. The change
into a + b (^) ⋅ X can be interpreted as observing the same X, however, mea-
sured in a different scale. In particular, if a and b are such that the mean and
variance of the resulting a + b (^) ⋅ X are 0 and 1, respectively, then a + b (^) ⋅ X is
called the standardization of X.
Property 2, stability under summation, ensures that the sum of an arbi-
trary number n of normal random variables, X 1 , X 2 ,... , Xn is, again, nor-
mally distributed provided that the random variables behave independently
of each other. This is important for aggregating quantities.
Furthermore, the normal distribution is often mentioned in the context
of the central limit theorem. It states that a sum of n random variables with
finite variance and identical distributions and being independent of each
other, converges in distribution to a normal random variable.^2 We restate
this formally as follows:
Let X 1 , X 2 ,... , Xn be identically distributed random variables with
mean E(Xi) = μ and var(Xi) = σ^2 and do not influence the outcome of each
other (i.e., are independent). Then, we have
Xn
n

N

i

i

n

D

−

→

⋅

=

∑ μ

σ

(^1) (,) 01 (B.2)
as the number n approaches infinity. The D above the convergence arrow in
equation (B.2) indicates that the distribution function of the left expression
convergences to the standard normal distribution.
Generally, for n = 30 in equation (B.2), we consider equality of the dis-
tributions; that is, the left-hand side is N(0,1) distributed. In certain cases,
depending on the distribution of the Xi and the corresponding parameter
(^2) There exist generalizations such that the distributions need no longer be identical.
However, this is beyond the scope of this appendix.