Stat310       Bivariate distributions


                           Hadley Wickham
Monday, 16 February 2009
1. Recap
               2. Transformations, the cdf and
                  the uniform distribution
               3. Intro...
Recap

                   X ~ Exponential(θ). Y = log(X).
                   What is fY(y)?
                   X ~ Uniform...
Theorem 3.5-1
                   IF
                   Y ~ Uniform(0, 1)
                   F a cdf
                   X =...
Theorem 3.5-2
                   IF
                   X has cdf F
                   Y = F(X)
                   THEN
   ...
http://www.johndcook.com/
          distribution_chart.html



Monday, 16 February 2009
Bivariate random
                               variables
                             Bivariate = two variables


Monday,...
Bivariate rv

                   Previously dealt with single random
                   variables at a time.
             ...
Monday, 16 February 2009
Monday, 16 February 2009
1
        f (x, y) =                          − 2 < x, y < 2
                   16
                                       ...
f (x, y) = c a < x, y < b


                           Is this a pdf?
                           How could we work out c?
...
f (x, y) ≥ 0    ∀x, y


                                    f (x, y) = 1
                               R2




Monday, 16 ...
S = {(x, y) : f (x, y) > 0}
      Called the support or
      sample space




Monday, 16 February 2009
What is the
  bivariate cdf going
  to look like?




Monday, 16 February 2009
What is the
  bivariate cdf going
  to look like?
                           x    y
 F (x, y) =                          f...
Your turn

                   F(x, y) =   c(x 2   +   y 2)   -1 < x, y < 1
                   What is c?
                 ...
Marginal distribution of X


                       fX (x) =                 f (x, y)dy
                                  ...
Demo



Monday, 16 February 2009
Feedback



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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

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