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# 11 Bivariate

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### 11 Bivariate

1. 1. Stat310 Bivariate distributions Hadley Wickham Monday, 16 February 2009
2. 2. 1. Recap 2. Transformations, the cdf and the uniform distribution 3. Introduction to bivariate distributions 4. Properties of pdf. Marginal pdfs & expectation 5. Feedback Monday, 16 February 2009
3. 3. Recap X ~ Exponential(θ). Y = log(X). What is fY(y)? X ~ Uniform(0, 10). Y = X2. What is fY(y)? Monday, 16 February 2009
4. 4. Theorem 3.5-1 IF Y ~ Uniform(0, 1) F a cdf X = F-1(Y) THEN X has cdf F(x) (Assume F strictly increasing for simplicity of proof, not needed in general) Monday, 16 February 2009
5. 5. Theorem 3.5-2 IF X has cdf F Y = F(X) THEN Y ~ Uniform(0, 1) (Assume F strictly increasing for simplicity of proof, not needed in general) Monday, 16 February 2009
6. 6. http://www.johndcook.com/ distribution_chart.html Monday, 16 February 2009
7. 7. Bivariate random variables Bivariate = two variables Monday, 16 February 2009
8. 8. Bivariate rv Previously dealt with single random variables at a time. Now we’re going to look at two (probably related) at a time New tool: multivariate calculus Monday, 16 February 2009
9. 9. Monday, 16 February 2009
10. 10. Monday, 16 February 2009
11. 11. 1 f (x, y) = − 2 < x, y < 2 16 What would you call What is: this distribution? • P(X < 0) ? • Draw diagrams and P(X < 0 and Y < 0) ? use your intuition • P(Y > 1) ? • P(X > Y) ? • P(X2 + Y2 < 1) Monday, 16 February 2009
12. 12. f (x, y) = c a < x, y < b Is this a pdf? How could we work out c? Monday, 16 February 2009
13. 13. f (x, y) ≥ 0 ∀x, y f (x, y) = 1 R2 Monday, 16 February 2009
14. 14. S = {(x, y) : f (x, y) > 0} Called the support or sample space Monday, 16 February 2009
15. 15. What is the bivariate cdf going to look like? Monday, 16 February 2009
16. 16. What is the bivariate cdf going to look like? x y F (x, y) = f (u, v)dvdu −∞ −∞ Monday, 16 February 2009
17. 17. Your turn F(x, y) = c(x 2 + y 2) -1 < x, y < 1 What is c? What is f(x, y)? Monday, 16 February 2009
18. 18. Marginal distribution of X fX (x) = f (x, y)dy R Marginal distribution of Y fY (y) = f (x, y)dx R Monday, 16 February 2009
19. 19. Demo Monday, 16 February 2009
20. 20. Feedback Monday, 16 February 2009