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# 11 Bivariate

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### Transcript

• 1. Stat310 Bivariate distributions Hadley Wickham Monday, 16 February 2009
• 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. 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. 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. 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. http://www.johndcook.com/ distribution_chart.html Monday, 16 February 2009
• 7. Bivariate random variables Bivariate = two variables Monday, 16 February 2009
• 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. Monday, 16 February 2009
• 10. Monday, 16 February 2009
• 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. f (x, y) = c a < x, y < b Is this a pdf? How could we work out c? Monday, 16 February 2009
• 13. f (x, y) ≥ 0 ∀x, y f (x, y) = 1 R2 Monday, 16 February 2009
• 14. S = {(x, y) : f (x, y) > 0} Called the support or sample space Monday, 16 February 2009
• 15. What is the bivariate cdf going to look like? Monday, 16 February 2009
• 16. What is the bivariate cdf going to look like? x y F (x, y) = f (u, v)dvdu −∞ −∞ Monday, 16 February 2009
• 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. 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. Demo Monday, 16 February 2009
• 20. Feedback Monday, 16 February 2009