14 Bivariate Transformations
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  • 1. Stat 310 Bivariate Transformations Garrett Grolemund
  • 2. Pick up handout
  • 3. 1. Example 2. Bivariate transformations 3. Calculating probabilities 4. Distribution function technique
  • 4. Question Suppose the basket is at (25, 0). Devise a way to calculate each shot’s distance from the basket using X and Y.
  • 5. Polar Coordinates r = √((x - 25)2 2 + ) y Ө = tan-1 (y/x)
  • 6. Polar Coordinates r = √((x -25)2 2 + ) y Ө = tan-1 (y/(x – 25))
  • 7. Bivariate Transformations (Transformations that involve two random variables at a time)
  • 8. Transformed Data
  • 9. Your Turn Suppose you own a portfolio of stocks. Let X1 be the amount of money your portfolio earns today, X2 be the amount of money it earns tomorrow, and so on… How would you calculate U and V, where U is the amount of money you’ll make on your best day during the next week, and V is the amount you’ll make on your worst day?
  • 10. Calculating Probabilities
  • 11. What is the probability that max(X1, X2 , X3 , X4 , X5 , X6 , X7) ≤ $100 ? min(X1, X2 , X3 , X4 , X5 , X6 , X7) ≤ $ -100 ?
  • 12. Recall from the univariate case, we have two methods of calculating probabilities of transformed variables Distribution Change of function variable technique technique
  • 13. Distribution function technique
  • 14. Suppose the Xi are iid. Is this a reasonable assumption? Then, we can calculate Fv(a) by P(V ≤ a) = P(min(Xi) ≤ a)
  • 15. Suppose the Xi are iid. Is this a reasonable assumption? Then, we can calculate Fv(a) by P(V ≤ a) = P(min(Xi) ≤ a) = 1 – P(min(Xi) > a)
  • 16. Suppose the Xi are iid. Is this a reasonable assumption? Then, we can calculate Fv(a) by P(V ≤ a) = P(min(Xi) ≤ a) = 1 – P(min(Xi) > a) = 1 – P(all Xi > a)
  • 17. = 1 – [P(X1 > a, X2 > a, … X7 > a)]
  • 18. = 1 – [P(X1 > a, X2 > a, … X7 > a)] = 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
  • 19. = 1 – [P(X1 > a, X2 > a, … X7 > a)] = 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ] (Because the Xi are independent)
  • 20. = 1 – [P(X1 > a, X2 > a, … X7 > a)] = 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ] (Because the Xi are independent) = 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ]
  • 21. = 1 – [P(X1 > a, X2 > a, … X7 > a)] = 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ] (Because the Xi are independent) = 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ] (because the Xi are identically distributed)
  • 22. = 1 – [P(X1 > a, X2 > a, … X7 > a)] = 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ] (Because the Xi are independent) = 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ] (because the Xi are identically distributed) = 1 – [P(X1 > a) 7 ]
  • 23. = 1 – [P(X1 > a, X2 > a, … X7 > a)] = 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ] (Because the Xi are independent) = 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ] (because the Xi are identically distributed) = 1 – [P(X1 > a) 7 ] = 1 – [ (1 – P(X1 ≤ a) )7 ]
  • 24. = 1 – [P(X1 > a, X2 > a, … X7 > a)] = 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ] (Because the Xi are independent) = 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ] (because the Xi are identically distributed) = 1 – [P(X1 > a) 7 ] = 1 – [ (1 – P(X1 ≤ a) )7 ] = 1 – [ (1 – Fx(a) ) 7 ]
  • 25. So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ] We can find the density of V by differentiating: fv(a) = Fv(a)
  • 26. So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ] We can find the density of V by differentiating: fv(a) = Fv(a) = {1 – [ (1 – Fx(a) ) 7 ]}
  • 27. So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ] We can find the density of V by differentiating: fv(a) = Fv(a) = {1 – [ (1 – Fx(a) ) 7 ]} = -7(1 – Fx(a) ) 6 (1 - Fx(a))
  • 28. So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ] We can find the density of V by differentiating: fv(a) = Fv(a) = {1 – [ (1 – Fx(a) ) 7 ]} = -7(1 – Fx(a) ) 6 (1 - Fx(a)) = 7(1 – Fx(a) ) 6 fx(a)
  • 29. Your Turn Work through the handout to find FU(a) and fU(a).
  • 30. What if we wish to find the joint distribution FU,V(a,b)? U = max(X, Y) V = min(X, Y) P(U < 2, V < 5) = ?
  • 31. Probability as volume under a surface f(x,y) P(Set A) X Set A Y
  • 32. P(U < 2, V < 5) = P( max(X, Y) < 5 min(X, Y) > 2) f(x,y) P(Set A) X Set A Y 5 5 P(U < 2, V < 5) = ∫ ∫ fx,y (x,y) dx dy 2 2
  • 33. But… •Computing double integrals can be hard •Finding correct bounds can be hard r = √((x - 25)2 + y2 ) Ө = tan-1 (y/(x – 25))
  • 34. Next time: Change of Variables
  • 35. Read Section 3.4