1. Stat310 Central limit theorem
Hadley Wickham
Thursday, 19 March 2009

2. 1. Exam
2. Recap
3. Sums of normals cont.
4. Chi-square distribution
5. Central limit theorem
Thursday, 19 March 2009

3. Exam
Too soon?
Alternatives:
Take home over recess
Extra big homework
Add on to final (i.e. worth 30% instead of 20%)
Thursday, 19 March 2009

4. Recap
If X ~ Normal(50, 25), what is:
P(X < 50) ?
P(40 < X < 55) ?
P(X < x) = 0.8 ?
Thursday, 19 March 2009

5. Sums of normals
Let Xi ~ Normal(μi, σi 2), independent
Y = c1X1 + c2X2 + … + cnXn
What is the distribution of Y?
(What is the mean and variance of Y?)
How can we work it out?
Thursday, 19 March 2009

6. Example
If Z1, Z2, Z3 are independent standard
normals, what is the distribution of:
Z1 - Z3
Z1 + Z2 + Z3
(Z1 + Z2 + Z3)/3
Thursday, 19 March 2009

7. Data
If X1, X2, …, Xn are iid N(μ, σ2)
What is the distribution of the sample
average?
What is the distribution of the sample
variance?
How are they related?
Thursday, 19 March 2009

8. Another special
distribution
If X ~ Normal(μ, σ2), and
V = (X - μ)2 / σ2 = Z2
Then
V ~ χ2(1)
Thursday, 19 March 2009

9. Chi-squared
Skipped over it in Chapter 3
Special case of the gamma distribution,
when θ = 2 and α = r / 2 (r an integer)
Mean = r, Variance = 2r
r is called degrees of freedom
Thursday, 19 March 2009

10. http://en.wikipedia.org/wiki/Chi-square_distribution
Thursday, 19 March 2009

11. Sums
Let Z1, Z2, …, Zn be iid N(0, 1)
W= Z12 + Z2 + … + Zn
2 2
Then
W ~ χ2(n)
Thursday, 19 March 2009

12. Intro to CLT
Thursday, 19 March 2009

13. Casino simulation
Crude simulation of a game a casino.
Wheel of fortune - costs $10 to play, and
winnings are distributed like a binomial
with p = 0.2.
What do you average winnings look like if
you play it many times?
Thursday, 19 March 2009

14. 3.0
2.8
2.6
mean
2.4
2.2
50 100 150 200
n
Thursday, 19 March 2009

15. 4.0
3.5
3.0
mean
2.5
2.0
50 100 150 200
n
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16. 2.4
2.2
2.0
mean
1.8
1.6
50 100 150 200
n
Thursday, 19 March 2009

17. Thursday, 19 March 2009

18. Thursday, 19 March 2009

19. n=1
250
200
150
count
100
50
0
0 2 4 6 8
mean
Thursday, 19 March 2009

20. n=2
200
150
count
100
50
0
0 1 2 3 4 5
mean
Thursday, 19 March 2009

21. n=3
150
100
count
50
0
0 1 2 3 4
mean
Thursday, 19 March 2009

22. n=4
150
100
count
50
0
0 1 2 3 4
mean
Thursday, 19 March 2009

23. n=5
140
120
100
80
count
60
40
20
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
mean
Thursday, 19 March 2009

24. 120
n=10
100
80
count
60
40
20
0
1.0 1.5 2.0 2.5 3.0
mean
Thursday, 19 March 2009

25. 140 n=20
120
100
80
count
60
40
20
0
1.5 2.0 2.5
mean
Thursday, 19 March 2009

26. n=100
150
100
count
50
0
1.6 1.8 2.0 2.2 2.4
mean
Thursday, 19 March 2009

27. 250 n=200
200
150
count
100
50
0
1.7 1.8 1.9 2.0 2.1 2.2 2.3
mean
Thursday, 19 March 2009

28. Mathematically
If X1, X2, …, Xn, are iid, and
X¯n − µ
Wn = √
σ/ n
then
lim Wn = Z ∼ Normal(0, 1)
n→∞
Thursday, 19 March 2009

29. I know of scarcely anything so apt to impress the
imagination as the wonderful form of cosmic order
expressed by the “Law of Frequency of Error”. ... It
reigns with serenity and in complete self-effacement,
amidst the wildest confusion. The huger the mob, and
the greater the apparent anarchy, the more perfect is
its sway. It is the supreme law of Unreason. Whenever
a large sample of chaotic elements are taken in hand
and marshaled in the order of their magnitude, an
unsuspected and most beautiful form of regularity
proves to have been latent all along.
— Sir Francis Galton (Natural Inheritance, 1889)
Thursday, 19 March 2009