Introduction to Financial Econometrics
Chapter 2 Review of Random Variables and
Probability Distributions
Eric Zivot
Department of Economics, University of Washington
January 18, 2000
This version: February 21, 2001
1 Random Variables
We start with a basic de&nition of a random variable
De&nition 1
A Random variable
X
is a variable that can take on a given set of
values, called the sample space and denoted
S
X
, where the likelihood of the values
in
S
X
is determined by
X
&
s probability distribution function (pdf).
For example, consider the price of Microsoft stock next month. Since the price
of Microsoft stock next month is not known with certainty today, we can consider
it a random variable. The price next month must be positive and realistically it
can±
t get too large. Therefore the sample space is the set of positive real numbers
bounded above by some large number. It is an open question as to what is the
best characterization of the probability distribution of stock prices. The lognormal
distribution is one possibility
1
.
As another example, consider a one month investment in Microsoft stock. That
is, we buy 1 share of Microsoft stock today and plan to sell it next month. Then
the return on this investment is a random variable since we do not know its value
today with certainty. In contrast to prices, returns can be positive or negative and are
bounded from below by 100%. The normal distribution is often a good approximation
to the distribution of simple monthly returns and is a better approximation to the
distribution of continuously compounded monthly returns.
As a &nal example, consider a variable
X
de&ned to be equal to one if the monthly
price change on Microsoft stock is positive and is equal to zero if the price change
1
If
P
is a positive random variable such that
ln
P
is normally distributed the
P
has a lognormal
distribution. We will discuss this distribution is later chapters.
1
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View Full Documentis zero or negative. Here the sample space is trivially the set
{
0
,
1
}
.
If it is equally
likely that the monthly price change is positive or negative (including zero) then the
probability that
X
=1
or
X
=0
is
0
.
5
.
1.1 Discrete Random Variables
Consider a random variable generically denoted
X
anditsseto
fpo
ss
ib
leva
lueso
r
sample space denoted
S
X
.
De&nition 2
A discrete random variable
X
is one that can take on a &nite number
of
n
di
f
erent values
x
1
,x
2
,...,x
n
or, at most, an in&nite number of di
f
erent values
x
1
2
,....
De&nition 3
The pdf of a discrete random variable, denoted
p
(
x
)
,
is a function such
that
p
(
x
)=Pr(
X
=
x
)
.
The pdf must satisfy (i)
p
(
x
)
≥
0
for all
x
∈
S
X
;
(ii)
p
(
x
)=0
for all
x/
∈
S
X
; and (iii)
P
x
∈
S
X
p
(
x
)=1
.
As an example, let
X
denote the annual return on Microsoft stock over the next
year. We might hypothesize that the annual return will be in! uenced by the general
state of the economy. Consider &ve possible states of the economy: depression, reces
sion, normal, mild boom and major boom. A stock analyst might forecast di
f
erent
values of the return for each possible state. Hence
X
is a discrete random variable
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 Econometrics, Normal Distribution, Probability theory

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