Consider the sample mean:
∑
=
=
n
1
i
i
X
n
1
X
Earlier lecture notes stated the standard error of
X
as:
n
)
X
(
se
σ
=
Define the standardized random variable:
n
X
)
X
(
se
X
Z
σ
μ

=
μ

=
The
Central Limit Theorem
states that as
n
becomes ‘large’ the
distribution of
Z
approaches the standard normal distribution.
That is,
)
1
,
0
(
N
~
Z
.
Therefore, a random variable that can be viewed as the sum of a
‘large’ number of independently and identically distributed random
variables will tend to have a normal distribution.
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View Full DocumentEcon 325 – Chapter 6
13
A computer simulation can be used to demonstrate the Central Limit
Theorem.
Consider a random variable that follows the uniform distribution
over the interval
[0, 10] .
A graph of the probability density function is below.
1/10
10
5
0
Econ 325 – Chapter 6
14
A statistical result is that the average of two
(
n
=2)
independent
uniform random variables has a triangular shape for the probability
density function.
To show this result with a computer simulation, select outcomes from
two uniform random variables on the interval
[0, 10]
and calculate
the average of the two values.
Repeat this 1000 times.
A histogram gives a graph of the frequency distribution of the
sample means generated by the computer simulation.
This gives a rough picture of the probability density function of the
sample mean.
The histogram generated by the computer simulation is shown
below.
The triangular shape of the theoretical probability density
function has also been sketched.
10
5
0
Econ 325 – Chapter 6
15
Now consider a sample size of
n
=25.
By the Central Limit Theorem it may be reasonable to assume that
the sample mean follows a normal distribution.
To start the computer simulation, select outcomes from 25 uniform
random variables on the interval
[0, 10]
and calculate the average of
the values.
Repeat this 1000 times.
The histogram generated by the computer simulation is shown
below.
A normal probability density function is also sketched to
reveal that, with
n
=25, the distribution of the sample mean is closely
approximated by the normal distribution.
6
5
4
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 Fall '10
 WHISTLER
 Normal Distribution, Variance, Probability theory

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