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20 Bivariate

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20 Bivariate

1. 1. Stat310 CLT, Bivariate Hadley Wickham Tuesday, 24 March 2009
2. 2. 1. Help session / Photos 2. Recap 3. Finish off CLT proof 4. Some animations 5. Bivariate normal distribution Tuesday, 24 March 2009
3. 3. Help session Changes: 5-6pm. Soyeon, not me Same place, DH 1049. Wednesday Photographer on Thursday Tuesday, 24 March 2009
4. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 23. Bivariate normal Our ﬁrst named bivariate distribution Tuesday, 24 March 2009
24. 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. 25. http://lstat.kuleuven.be/ java/version2.0/ Applet030.html Tuesday, 24 March 2009
26. 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. 27. Independence If ρ = 0, what does that imply about X and Y? Tuesday, 24 March 2009
28. 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