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Hadley Wickham
Stat310CLT, Bivariate
Tuesday, 24 March 2009
1. Help session / Photos
2. Recap
3. Finish off CLT proof
4. Some animations
5. Bivariate normal distribution
Tuesday, 24 March 2009
Changes: 5-6pm. Soyeon, not me
Same place, DH 1049. Wednesday
Photographer on Thursday
Help session
Tuesday, 24 March 2009
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 254:00 - 5:30 pm
Brochstein Pavilion
Tuesday, 24 March 2009
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
Mathematically
Wn =X̄n ! µ
!/"
n
limn!"
Wn = Z ! Normal(0, 1)
If X1, X2, …, Xn, are iid, and
then
Tuesday, 24 March 2009
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
Alternative expressions
!n(X̄n " µ) D# N(0, !2)
limn!"
P (Wn < z) = !(z)
WnD! N(0, 1)
Tuesday, 24 March 2009
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
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
mean
count 0
100
200
300
400
0
100
200
300
400
1
3
0.0 0.2 0.4 0.6 0.8 1.0
2
4
0.0 0.2 0.4 0.6 0.8 1.0
Tuesday, 24 March 2009
mean
coun
t 0
200
400
600
0
200
400
600
5
20
0.0 0.2 0.4 0.6 0.8 1.0
10
50
0.0 0.2 0.4 0.6 0.8 1.0
Tuesday, 24 March 2009
(mean − 0.5) * sqrt(n)/sqrt(1/8)
coun
t 0
100
200
300
400
0
100
200
300
400
1
3
−4 −2 0 2 4
2
4
−4 −2 0 2 4
Standardise
Tuesday, 24 March 2009
(mean − 0.5) * sqrt(n)/sqrt(1/8)
coun
t 0
50
100
150
200
0
50
100
150
200
5
20
−4 −2 0 2 4
10
50
−4 −2 0 2 4
Tuesday, 24 March 2009
mean
count 0
50
100
150
200
0
50
100
150
200
1
3
−4 −2 0 2 4
2
4
−4 −2 0 2 4
Calibration5000 standard normals
Tuesday, 24 March 2009
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
mean
coun
t 0
500
1000
1500
0
500
1000
1500
1
10
−1.0 −0.5 0.0 0.5 1.0
5
50
−1.0 −0.5 0.0 0.5 1.0
Tuesday, 24 March 2009
mean
count 0
200
400
600
800
0
200
400
600
800
100
200
−1.0 −0.5 0.0 0.5 1.0
150
250
−1.0 −0.5 0.0 0.5 1.0
Tuesday, 24 March 2009
(mean − rlt_mean) * sqrt(n)/sqrt(rlt_var)
coun
t 0
500
1000
1500
0
500
1000
1500
1
10
−4 −2 0 2 4
5
50
−4 −2 0 2 4
Standardise
Tuesday, 24 March 2009
(mean − rlt_mean) * sqrt(n)/sqrt(rlt_var)
coun
t 0
100
200
300
0
100
200
300
100
200
−4 −2 0 2 4
150
250
−4 −2 0 2 4
Tuesday, 24 March 2009
(mean − rlt_mean) * sqrt(n)/sqrt(rlt_var)
coun
t 0
50
100
150
200
250
0
50
100
150
200
250
300
600
−4 −2 0 2 4
400
800
−4 −2 0 2 4
Tuesday, 24 March 2009
mean
count 0
50
100
150
200
250
0
50
100
150
200
250
1
3
−4 −2 0 2 4
2
4
−4 −2 0 2 4
Calibration3000 standard normals
Tuesday, 24 March 2009
Bivariate normalOur first named bivariate distribution
Tuesday, 24 March 2009
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
http://lstat.kuleuven.be/java/version2.0/Applet030.html
Tuesday, 24 March 2009
f(x, y) =1
2!"x"y
!1! #2
exp"!q(x, y)
2
#
q(x, y) =1
1! !2
!z2x + z2
y ! 2!zxzy
"
zx =x! µx
!xzy =
x! µy
!y
Tuesday, 24 March 2009
Independence
If ρ = 0, what does that imply about X and Y?
Tuesday, 24 March 2009
Marginal and conditionals
Both marginal and conditional distributions are normal.
Y ! Normal(µy, !2y)X ! Normal(µx, !2
x)
X|Y ! Normal(µx + !"x
"y(y " µy), "2
x(1" !2))
Y|X ! Normal(µy + !"y
"x(x" µx), "2
y(1" !2))
Tuesday, 24 March 2009
