Lecture 2 Notes ,
Central Limit Theorem
– Certain statistics (sample means, for example)
are normally
distributed, If the distribution is symmetrical, a small sample the larger the sample size
the more normal the distribution will become.
An Exercise to demonstrate the Central Limit Theorem
:
X
P(X)
(X  µ
x
)
2
•P(Xi)
P()
1
1/3
4/3
3
1/3
0
3/9
5
1/3
4/3
2/9
µ= Σ (Xi•P(Xi)) = 3
σ
2
= E(Xµ
x
)
2
= Σ (X  µ
x
)
2
•P(Xi) = 8/3
1/9
Sample Size 2
1
2
3
4
5
Notice that with a sample size of only two, the distribution of sample means can be
approximated by the normal distribution. That is to say, sample means quickly take on
the shape of the normal even when the sample size is very small. Notice, however, that
the original distribution of the variable was symmetrical. If the underlying distribution is
highly skewed, then it would take a much larger sample size for the distribution of
sample means to become normal.
P()
(P)
(  µ)
2
(  µ)
2
•P()
1, 1
1
1
1/9
1/9
4
4/9
1, 3
2
2
2/9
4/9
1
2/9
1, 5
3
3
3/9
9/9
0
0
3, 1
2
4
2/9
8/9
1
2/9
3, 3
3
5
1/9
5/9
4
4/9
3, 5
4
µ = 3
σ
2
= 4/3
5, 1
3
5, 3
4
5, 5
5
σ
2
= Σ ( µ
)
2
•
P()
σ
2
= (8/3)/2 = 4/3
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
E() = µ
x
(the population mean, regardless of population size)
E() = Σ (Xi•P(Xi)) and Σ •P() = µ
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '10
 chandrasekhar
 Statistics, Central Limit Theorem, Normal Distribution

Click to edit the document details