1. Stat310 CLT, Bivariate
Hadley Wickham
Tuesday, 24 March 2009

2. 1. Help session / Photos
2. Recap
3. Finish off CLT proof
4. Some animations
5. Bivariate normal distribution
Tuesday, 24 March 2009

3. Help session
Changes: 5-6pm. Soyeon, not me
Same place, DH 1049. Wednesday
Photographer on Thursday
Tuesday, 24 March 2009

4. VIGRE Poster session
VIGRE is a program sponsored by the National Science
Foundation to carry out innovative educational programs
in which research and education are integrated and in
which undergraduates, graduate students, postdoctoral
fellows, and faculty are mutually supportive.
Wednesday, March 25
4:00 - 5:30 pm
Brochstein Pavilion
Tuesday, 24 March 2009

5. Recap
In your own words (or pictures or
symbols) write down what the central limit
theorem means
(I’ll collect these this time, so please use a
sheet of paper)
Tuesday, 24 March 2009

6. Mathematically
If X1, X2, …, Xn, are iid, and
¯n − µ
X
Wn = √
σ/ n
then
lim Wn = Z ∼ Normal(0, 1)
n→∞
Tuesday, 24 March 2009

7. Fuller proof
If we want to be completely correct,
we’ve missed a few important proofs:
If a series of mgf’s converges to a
function, does the cdf/pdf also converge?
Is the error term really small enough?
See section 5.7 or the pdf linked from the
website for more of these details.
Tuesday, 24 March 2009

8. Alternative expressions
D
Wn → N (0, 1)
√
¯ n − µ) → N (0, σ 2 )
D
n(X
lim P (Wn < z) = Φ(z)
n→∞
Tuesday, 24 March 2009

9. 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)
Tuesday, 24 March 2009

10. Why is it useful?
Many types of averages:
Average number of deaths per month
Cases of cancer per state
A couple more illustrations
Tuesday, 24 March 2009

11. 1 2
400
300
200
100
0
count
3 4
400
300
200
100
0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
mean
Tuesday, 24 March 2009

12. 5 10
600
400
200
0
count
20 50
600
400
200
0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
mean
Tuesday, 24 March 2009

13. 1 2
400
300
200
100
0
count
Standardise
3 4
400
300
200
100
0
−4 −2 0 2 4 −4 −2 0 2 4
(mean − 0.5) * sqrt(n)/sqrt(1/8)
Tuesday, 24 March 2009

14. 5 10
200
150
100
50
0
count
20 50
200
150
100
50
0
−4 −2 0 2 4 −4 −2 0 2 4
(mean − 0.5) * sqrt(n)/sqrt(1/8)
Tuesday, 24 March 2009

15. 1 2
200
150
100
50
Calibration
0
count
3 4
5000 standard normals
200
150
100
50
0
−4 −2 0 2 4 −4 −2 0 2 4
mean
Tuesday, 24 March 2009

16. Counterexample
Playing roulette at a casino, betting 1
dollar on red. What is the distribution of
my average winnings?
Probability of winning $1: 18/38
Probability of losing $1: 20/38
Tuesday, 24 March 2009

17. 1 5
1500
1000
500
0
count
10 50
1500
1000
500
0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
mean
Tuesday, 24 March 2009

18. 100 150
800
600
400
200
0
count
200 250
800
600
400
200
0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
mean
Tuesday, 24 March 2009

19. 1 5
1500
1000
500
0
count
Standardise
10 50
1500
1000
500
0
−4 −2 0 2 4 −4 −2 0 2 4
(mean − rlt_mean) * sqrt(n)/sqrt(rlt_var)
Tuesday, 24 March 2009

20. 100 150
300
200
100
0
count
200 250
300
200
100
0
−4 −2 0 2 4 −4 −2 0 2 4
(mean − rlt_mean) * sqrt(n)/sqrt(rlt_var)
Tuesday, 24 March 2009

21. 300 400
250
200
150
100
50
0
count
600 800
250
200
150
100
50
0
−4 −2 0 2 4 −4 −2 0 2 4
(mean − rlt_mean) * sqrt(n)/sqrt(rlt_var)
Tuesday, 24 March 2009

22. 1 2
250
200
150
100
50
Calibration
0
count
3 4
3000 standard normals
250
200
150
100
50
0
−4 −2 0 2 4 −4 −2 0 2 4
mean
Tuesday, 24 March 2009

23. Bivariate normal
Our first named bivariate distribution
Tuesday, 24 March 2009

24. Bivariate Normal
A bivariate distribution where all marginal
and conditional distributions are normal.
Five parameters: two means, two
variances, and correlation
Tuesday, 24 March 2009

25. http://lstat.kuleuven.be/
java/version2.0/
Applet030.html
Tuesday, 24 March 2009

26. 1 q(x, y)
f (x, y) = exp −
2
2πσx σy 1−ρ 2
1
q(x, y) = zx + zy − 2ρzx zy
2 2
1−ρ 2
x − µx x − µy
zx = zy =
σy
σx
Tuesday, 24 March 2009

27. Independence
If ρ = 0, what does that imply about X and
Y?
Tuesday, 24 March 2009

28. Marginal and
conditionals
Both marginal and conditional
distributions are normal.
X∼ Normal(µx , σx )
Y∼ Normal(µy , σy )
2 2
σx
X|Y ∼ Normal(µx + ρ (y − µy ), σx (1 − ρ ))
2 2
σy
σy
Y|X ∼ Normal(µy + ρ (x − µx ), σy (1 − ρ ))
2 2
σx
Tuesday, 24 March 2009