1
Some Formulas neglected in Anderson, Sweeny, and Williams,
with a Digression on Statistics and Finance
Transformations of a Single Random variable:
If you have a case where a new random variable is defined as a linear
transformation of a another random variable, there is an easy way to get the mean and
variance of the new random variable.
Assume you are given
Y
ab
X
=+
, where
a
and
b
are constants, and
Y
is a new random variable being defined as this linear transformation
of the random variable
X
.
If we know the mean and variance of the random variable
X
, we can easily find the mean and variance of the random variable
Y
using the
following two formulas.
µ
σσ
yx
b
=
22
2
.
A word of warning: this shortcut only works if
Y
is a
linear
transformation of
X
.
For instance, if
2
YX
=
it is not true
that
2
µµ
=
.
If
2
=
(or for any nonlinear
transformation) we must use the equation that
Egx
gxf x
x
bg
ch
bgbg
=
∑
.
An Example of a NonLinear Transformation
Suppose the random variable
X
has the following distribution:
x
f(x)
0
.343
1
.441
2
.189
3
.027
It is easily verified (we did the example in class) that the mean of the random
variable is
() (
)
3
0
0.9
x
EX
x
fx
=
==
∑
.
Suppose
()
2
gx x
=
.
One might expect that
() ()
2
0.9
0.81
Ex
=
 However, this is
WRONG
.
The correct calculation is
x
x
2
f(x)
x
2
f(x)
0
0
.343
0
1
1
.441
.441
2
4
.189
.756
3
9
.027
.243
2
xf x
∑
1.440
Why is this important?
It is central in the economics of uncertainty.
When an
agent’s utility depends on an uncertain outcome (a very common case in the real world)
the utility achieved becomes random, so the agent can’t be thought of as maximizing
utility.
The simpliest generalization of utility theory that can cover this case is to assume
that under uncertainty, agents maximize
expected utility
.
If utility,
U
, depends on a
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random variable
X
– which for concreteness you can think of as wealth – then the agent
maximizes
()
()()
EU x
U xf x
=
∑
.
The fact that
Egx
gEx
≠
has important
implications in economic theory.
For instance, it implies that when wealth is uncertain,
an expected utility maximizer may not wish to select a strategy that maximizes expected
wealth.
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 Spring '08
 Staff
 Standard Deviation, Variance, Utility, Probability theory, Ω

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