**
An Introduction to
Random
Variables**

■
What
is a
random variable?

■
What
is a
discrete
random
variable?

■
What
are
the
mean, variance,
and standard
deviation
of a
random
variable?

■
What
is a
continuous
random variable?

■
What
is a
probability
density
function?

■
What
are
independent
random
variables?

In
today’s
world,
the
only
thing
that’s
certain
is
that
we
face a
great
deal of
uncertainty.
In
the
next nine
chapters,
I’ll give
you
some
powerful
techniques
that
you
can
use
to incorporate
uncertainty
in business
models. The
key
building block
in modeling
uncertainty
is
under-
standing how
to use random
variables.

What is a random
variable?

Any
situation
whose
outcome
is uncertain
is called
an *
experiment*.
The
value
of a
random
variable is based on
the
(uncertain)
outcome of
an experiment.
For example,
tossing
a pair
of
dice
is an
experiment,
and a
random
variable
might be
defined
as
the
sum of
the
values
shown
on each
die. In
this
case,
the
random
variable
could
assume any
of
the
val- ues
2, 3,
and so
on up to
12.
As another example,
consider
the
experiment
of selling a
new
video
game
console,
for
which a
random
variable
might be
defined
as
the
market
share
for
this
new product.

What is a discrete
random
variable?

A
random
variable
is
discrete
if
it
can
assume
a
finite
number
of
possible
values.
Here
are
some
examples
of discrete
random
variables:

❑
Number
of
potential
competitors
for
your product

❑
Number
of
aces drawn
in a
five-card
poker
hand

❑
Number
of
car
accidents
you
have (hopefully
zero!)
in a
year

❑
Number
of dots
showing
on a die

❑
Number
of free
throws
out of 12
that
Steve
Nash
makes
during
a
basketball
game

What are
the mean,
variance, and
standard
deviation
of a
random
variable?

In Chapter 37,
“Summarizing
Data with
Descriptive Statistics,”
I discussed
the
mean, variance,
and
standard
deviation
for
a
data
set.
In
essence,
the
mean
of a
random
variable

(often denoted
by
µ) is the
average
value of
the
random variable
we
would
expect if
we
performed
an
experiment
many
times.
The
mean
of
a random
variable
is
often
referred

to
as
the
random variable’s
*expected
*value.
The variance
of a random
variable (often
denoted
by
σ2)
is
the
average
value
of
the
squared
deviation from
the
mean of
a random
variable
that
we
would
expect if
we
performed
our
experiment
many
times. The
stan-

dard
deviation
of
a
random
variable
(often
denoted
by
σ)
is simply
the
square
root
of its variance.
As with
data sets,
the
mean
of a random variable
is a
summary
measure
for
a typical value of
the
random variable,
whereas
the
variance
and
standard
deviation
mea-
sure
the
spread
of
the
random
variable
about
its mean.

As an example
of how
to compute
the mean,
variance,
and standard
deviation
of a ran-
dom variable, suppose
we believe
that
the
return
on the
stock
market during
the
next year
is governed
by
the
following
probabilities:

**Probability
Market return**

.40
+20 percent

.30
0 percent

.30
-20 percent

Hand calculations show
the
following:

µ=.40*(.20)+.30*(.00)+.30*(–.20)=.02
or
2
percent

σ2=.4*(.20–.02)2+.30*(.0–.02)2+.30*(–.20–.02)2=.0276

Then
σ*=.166
*
or
16.6
percent.

In
the
file
Meanvariance.xlsx
(shown
in
Figure
55-1),
I’ve
verified
these
computations.

Figure
55-1
Computing
the mean,
standard
deviation,
and variance
of a random
variable

I computed
the
mean
of our
market
return
in
cell C9 with
the
formula

=*SUMPRODUCT(B4:B6,C4:C6)*.
This
formula
multiplies each value
of
the
random
variable
by
its probability
and sums up
the
products.

To
compute
the
variance
of our
market
return,
I determined
the
squared
deviation
of each value
of
the
random variable
from
its
mean
by
copying from
D4
to
D5:D6
the
formula
=*(B4–$C$9)^2*.
Then,
in
cell
C10,
I computed
the
variance
of
the
market
return

as
the
average
squared
deviation from
the
mean with
the
formula
=*SUMPRODUCT*

*
(C4:C6,D4:D6)*.
Finally, I
computed
the
standard
deviation of
the
market
return
in cell

C11 with
the
formula
=*SQ**RT(C10)*.

What is a continuous
random
variable?

A
continuous
random
variable
is a
random
variable
that
can
assume
a
very
large
number or,
to
all
intents and
purposes, an
infinite
number
of values.
Here
are
some
examples
of continuous random
variables:

❑
Price
of
Microsoft
stock
one
year
from
now

❑
Market
share
for a
new
product

❑
Market
size for
a
new
product

❑
Cost
of developing
a new
product

❑
Newborn
baby’s
weight

❑
Person’s
IQ

❑
Dirk
Nowitzki’s
three-point
shooting
percentage
during
next
season

What is a probability
density
function?

A discrete
random
variable
can
be specified
by
a list
of values
and the probability
of occurrence
for
each
value
of
the
random
variable.
Because
a
continuous
random
variable can
assume
an infinite
number
of values, we
can’t list the
probability
of occurrence
for each
value of a
continuous
random variable.
A continuous
random variable
is com- pletely
described by
its *probability
density
function*.
For
example,
the
probability
density function
for
a randomly
chosen
person’s IQ
is shown
in Figure
55-2.

Figure
55-2
Probability
density function for IQs

A
probability
density
function
(pdf) has
the
following
properties:

❑
The
value of
the
pdf is
always
greater
than or
equal
to 0.

❑
The
area
under
the
pdf
equals
1.

❑
The
height
of
the
density
function
for
a
value
*
x
*
of
a
random
variable
is
proportional

to
the
likelihood
that
the
random
variable
assumes
a
value
near
*
x*.
For
example,
the
height
of
the
density
for
an
IQ
of
83
is
roughly
half
the
height
of
the
density
for
an
IQ

of
100.
This
tells
us
that
IQs
near
83 are
approximately
half
as
likely
as
IQs
around

100.
Also,
because
the
density
peaks
at
100,
IQs
around
100
are
most
likely.

❑
The
probability
that
a
continuous
random
variable
assumes
a
range
of
values
equals
the
corresponding
area
under
the
density
function.
For
example,
the
frac- tion
of people
having
IQs from
80
through
100
is simply
the
area
under
the
density from
80 through
100.

What are
independent random
variables?

A
set
of
random
variables
are
independent
if knowledge
of
the
value
of any
of
their
sub- sets
tells
you
nothing
about
the
values
of
the
other
random variables.
For
example,
the
number
of
games
won
by
the
Indiana
University
football
team during
a year
is indepen-
dent of
the
percentage
return
on Microsoft
during
the
same
year.
Knowing
that
Indiana did
very
well
would not
change
your
view
of how
Microsoft
stock did
during
the
year.

On
the
other
hand,
the
return
on Microsoft
stock
and
Intel stock
are
not independent.

If we
are
told
that
Microsoft
stock had
a high
return
in one
year,
in all
likelihood,
com- puter sales
were
high,
which
tells
us
that
Intel probably
had a
good
year
as
well.

**
Problems**

**
1.
**Identify
the
following
random
variables
as discrete
or continuous:

❑
Number
of
games
Kerry
Wood
wins
for
the
Chicago
Cubs
next
season

❑
Number
that comes
up when spinning a
roulette
wheel

❑
Unit
sales of
Tablet
PCs next
year

❑
Length
of time that a
light
bulb lasts
before
it burns
out

**
2. **Compute
the
mean, variance,
and standard
deviation
of
the
number of dots
showing
when a
die is
tossed.

**
3. **Determine
whether
the
following
random variables
are independent:

❑
Daily
temperature
and
sales
at
an
ice cream
store

❑
Suit
and
number
of a card
drawn
from
a deck
of playing
cards

❑
Inflation
and
return
on the
stock
market

❑
Price
charged
for
and
the
number of
units
sold of
a car

**
4. **The
current
price
of a company’s
stock
is
$20.
The
company
is a
takeover
target.
If
the
takeover
is successful,
the
company’s
stock
price will increase
to
$30. If the
takeover
is unsuccessful,
the
stock
price will drop
to
$12.
Determine
the
range
of values for
the
probability
of a
successful
takeover
that
would
make
it
worthwhile
to
purchase
the
stock
today.
Assume
your
goal
is
to
maximize
your
expected
profit.
Hint:
Use
the
Microsoft
Office
Excel
2007
Goal
Seek
command,
which
is
discussed
in
detail
in Chapter
16,
“The
Goal Seek
Command.”