12 Bivariate

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12 Bivariate

  1. 1. Stat310 Bivariate random variables Hadley Wickham Friday, 26 February 2010
  2. 2. Assessment • Please pick up any homework you haven’t got already. • Will be grading tests tomorrow to get back to you on Thursday • Drop deadline is Feb 26 – if you are thinking of dropping and would like an interim grade, email me Friday morning Friday, 26 February 2010
  3. 3. 1. Introduction to bivariate random variables 2. The important bits of multivariate calculus 3. Independence Friday, 26 February 2010
  4. 4. Bivariate rv Previously dealt with one random variable at a time. Now we’re going to look at two (probably related) at a time. A random experiment where we measure two things (not just one). New tool: multivariate calculus Friday, 26 February 2010
  5. 5. Friday, 26 February 2010
  6. 6. Friday, 26 February 2010
  7. 7. 1 f (x, y) = − 2 < x, y < 2 16 What is: What would you call this distribution? • P(X < 0) ? • P(X < 0 and Y < 0) ? Draw diagrams and • P(Y > 1) ? use your intuition • P(X > Y) ? • P(X2 + Y2 < 1) Friday, 26 February 2010
  8. 8. f (x, y) = c a < x, y < b Is this a pdf? How could we work out c? Friday, 26 February 2010
  9. 9. Your turn Given what you know about univariate pdfs and pmfs, guess the conditions that a bivariate function must satisfy to be a bivariate pdf/pmf. Friday, 26 February 2010
  10. 10. pdf ∞ ∞ −∞ ∞ f (x, y) dy dx = 1 f (x, y) ≥ 0 pmf f (x, y) = 1 f (x, y) ≥ 0 x,y Friday, 26 February 2010
  11. 11. S = {(x, y) : f (x, y) 0} The support or sample space Friday, 26 February 2010
  12. 12. P (a X b, c Y d) = d b f (x, y) dx dy c a Friday, 26 February 2010
  13. 13. What is the cdf going to look like? P (X x, Y y) = Friday, 26 February 2010
  14. 14. What is the cdf going to look like? P (X x, Y y) = x y F (x, y) = f (u, v)dvdu −∞ −∞ Friday, 26 February 2010
  15. 15. Multivariate calculus Friday, 26 February 2010
  16. 16. Important bits Partial derivatives Multiple integrals (2d change of variable - after spring break) Use wolfram alpha. Wikipedia articles are decent. Friday, 26 February 2010
  17. 17. Your turn F(x, y) = c(x 2 + y 2) -1 x, y 1 What is c? What is f(x, y)? Friday, 26 February 2010
  18. 18. Marginal distributions fX (x) = f (x, y)dy R fY (y) = f (x, y)dx R Friday, 26 February 2010
  19. 19. Independence How can we tell if two random variables are independent? Need to go back to our definition. Friday, 26 February 2010
  20. 20. Dependence Only one way for rv’s to be independent. Many ways to be dependent. Useful to have some measurements to summarise common forms of dependence. Next time we’ll use one you’ve hopefully heard of before: correlation, a measurement of linear dependence. Friday, 26 February 2010
  21. 21. Read 3.3 and 3.3.1 Friday, 26 February 2010
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