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  • 1. Stat310 Bivariate distributions Hadley Wickham Friday, 26 February 2010
  • 2. 1. Test 2. Example 3. CDF 4. Expectation 5. Conditional distributions 6. Correlation Friday, 26 February 2010
  • 3. Test Friday, 26 February 2010
  • 4. 15 10 n 5 C B A 0 10 15 20 25 30 35 40 Grade Friday, 26 February 2010
  • 5. But, you can expect a boost of about 10% from your homeworks 15 10 n 5 C B A 0 10 15 20 25 30 35 40 Grade Friday, 26 February 2010
  • 6. Test It was a bit harder than I intended (Question 1 was supposed to be the easiest ☹). I don’t curve, but I will provide opportunities for you to improve your overall grade. Extra credit this week worth 5 points - you may redo one question from the exam. Model answers will be posted after spring break. Friday, 26 February 2010
  • 7. Example Friday, 26 February 2010
  • 8. ● ● ● ● 70 ● ● ● ● ● ● 60 ● 50 type ● ● ● 3pt ● ● ● ● driving layup ● 40 ● hook y ● ● ● ● ● jump ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● layup 30 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● turnaround jump ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 10 20 30 40 50 x Friday, 26 February 2010
  • 9. 3pt driving layup hook ● ● ● ● 70 ● ● ● ● ● ● 60 ● 50 ● ● ● ● ● ● 40 ● ● ●●● ●● ● ● ● ●●●●●●●●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●● ●●●●●●●● ● 30 ●●● ●●●●● ●●●● ● ●● ●●●●●●● ● ● ● ●●●●●● ●●●●●● ●●●●● ●●● ●●● ●●●●●● ●●●●● ●●●●● ●●●●● ●●●● ●●●● ●●● ●●● ●● ●●●● ●●●● ●●●● ●●●● ●●● ●● ●●● ●● ●●● ●●● ●●● ●●● ●●●● ● ●●●● ●●● 20 ●●● ●●●● ● ●●● ●●● ●● ●● ●● ●● ●● ● ●● ●● ● ● ● ● ● ●● ●●● ●●●●●●● ●● ● ● ●● ● ●● ● ●●●● ● ●● ● ●●●●● ● ● ●● ● ●● ● ●● ●● ● ● ● ●● ● ● ●●●●●●● ● ● 10 ●● ●● ●● ● ● ●● ●● ●●● ● ● ●●●●●●●● ●●● ●●●●●● ●●●●●● ●● ●● ● ●● ● ● ●● ● ●●●● ●●● ●●● ●● ● ●●●● ●●● ● ● ●●● ● ●● ●● ● ● ● ● ●● ●● ●● ●● ●● ● ●● ● ● ●● ● ● ● ● ●●● ● ● ● ● ●● ●● ●● ● jump layup turnaround jump y 70 60 50 40 30 ● ● ● ● ●●●●●●●● ●●●● ● ●●●●●●●●●●● ● ●●●●●●●●●●●● ●●●●●●●●● ● ●●●●●●●● ● ●●● ● ●●●●●●●●●●●●●●● ●●●●●● ● ●● ● ●●●●●●●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●● ● ● 20 ●●●●●● ● ●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●● ● ●●●●● ●● ●●●● ● ● ●● ●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●● ●●●●●●● ●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●● ●● ●●●●● ● ● ●● ● ● ● ●● ● ● ●●●●●●●●●●●●●●●●●●●●● ●● ●●● ●● ●● ●●●●●●●● ●●●● ●●●●●●● ● ● ●● ● ● ●●●●●● ● ● ● ● ● ●●●● ● ● ● ●●●●●●●●●●●●●●●●●●● ● ●●● ● ● ●●●●●●●●●● ● ● ● ● ●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ● ● ●●● ●●●●●●●● ● ●●●● ● ●●● ● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ● ●● ● ● ● ● ●●●●● ●● ●● ● ● 10 ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●● ●●●● ● ●●● ●●●●●●●●● ●●●●● ●●●●● ●●●●●● ●●●● ●●●● ●●●●●● ●●●● ●●●●● ●●● ●●● ●● ●● ● ● ● ● ● ●●●●● ●●● ● ●●●●● ●● ● ● ●●●● ●●●●●●●●●●● ●●●●●●● ● ●●●●●●●●●● ●●●●●●●●●● ●●●● ●●● ●● ● ●●● ●●●●●●●●● ● ●●●●● ● ● ● ●●● ●●● ● ●●●●● ● ●●● ● ● ●●●●●●●●● ●●●●●●●●● ●●●●●●●● ● ●●●●● ●● ●● ●● ●●●● ●●●● ● ●● ●●● ● ● ●● ●● ●● ●●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 x Friday, 26 February 2010
  • 10. 3pt driving layup hook ● ● ● ● 70 ● ● ● ● ● ● 60 ● 50 ● ● ● ● ● ● 40 ● ● ●●● ●● ● ● ● ●●●●●●●●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●● ●●●●●●●● ● 30 ●●● ●●●●● ●●●● ● ●● ●●●●●●● ● ● ● ●●●●●● ●●●●●● ●●●●● ●●● ●●● ●●●●●● ●●●●● ●●●●● ●●●●● ●●●● ●●●● ●●● ●●● ●● ●●●● ●●●● ●●●● ●●●● ●●● ●● ●●● ●● ●●● ●●● ●●● ●●● ●●●● ● ●●●● ●●● 20 ●●● ●●●● ● ●●● ●●● ●● ●● ●● ●● ●● ● ●● ●● ● ● ● ● ● ●● ●●● ●●●●●●● ●● ● ● ●● ● ●● ● ●●●● ● ●● ● ●●●●● ● ● ●● ● ●● ● ●● ●● ● ● ● ●● ● ● ●●●●●●● ● ● 10 ●● ●● ●● ● ● ●● ●● ●●● ● ● ●●●●●●●● ●●● ●●●●●● ●●●●●● ●● ●● ● ●● ● ● ●● ● ●●●● ●●● ●●● ●● ● ●●●● ●●● ● ● ●●● ● ●● ●● ● ● ● ● ●● ●● ●● ●● ●● ● ●● ● ● ●● ● ● ● ● ●●● ● ● ● ● ●● ●● ●● ● jump layup turnaround jump y 70 60 50 40 30 ● ● ● ● ●●●●●●●● ●●●● ● ●●●●●●●●●●● ● ●●●●●●●●●●●● ●●●●●●●●● ● ●●●●●●●● ● ●●● ● ●●●●●●●●●●●●●●● ●●●●●● ● ●● ● ●●●●●●●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●● ● ● 20 ●●●●●● ● ●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●● ● ●●●●● ●● ●●●● ● ● ●● ●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●● ●●●●●●● ●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●● ●● ●●●●● ● ● ●● ● ● ● ●● ● ● ●●●●●●●●●●●●●●●●●●●●● ●● ●●● ●● ●● ●●●●●●●● ●●●● ●●●●●●● ● ● ●● ● ● ●●●●●● ● ● ● ● ● ●●●● ● ● ● ●●●●●●●●●●●●●●●●●●● ● ●●● ● ● ●●●●●●●●●● ● ● ● ● ●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ● ● ●●● ●●●●●●●● ● ●●●● ● ●●● ● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ● ●● ● ● ● ● ●●●●● ●● ●● ● ● 10 ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●● ●●●● ● ●●● ●●●●●●●●● ●●●●● ●●●●● ●●●●●● ●●●● ●●●● ●●●●●● ●●●● ●●●●● ●●● ●●● ●● ●● ● ● ● ● ● ●●●●● ●●● ● ●●●●● ●● ● ● ●●●● ●●●●●●●●●●● ●●●●●●● ● ●●●●●●●●●● ●●●●●●●●●● ●●●● ●●● ●● ● ●●● ●●●●●●●●● ● ●●●●● ● ● ● ●●● ●●● ● ●●●●● ● ●●● ● ● ●●●●●●●●● ●●●●●●●●● ●●●●●●●● ● ●●●●● ●● ●● ●● ●●●● ●●●● ● ●● ●●● ● ● ●● ●● ●● ●●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 x Friday, 26 February 2010
  • 11. Marginal distribution of y 140 120 100 80 ..count.. 60 40 20 0 5 10 15 20 25 30 y Friday, 26 February 2010
  • 12. Marginal distribution of x 80 60 ..count.. 40 20 0 10 20 30 40 x Friday, 26 February 2010
  • 13. Joint distribution of x and y 2114 jumps 30 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● y 15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 20 30 40 x Friday, 26 February 2010
  • 14. ● 30 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● 25 ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ●● ● ● ●●● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●●● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ●●●●● ● ● ●●●●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●●●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● 20 ● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ●● ●● ● ● ● ●● ● ●●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ●●● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ●● ● ●● ● ●●● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ●●● ● ●● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ●● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● y ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● 15 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ●●●●● ●● ● ●● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ●●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●●● ●●● ● ●● ●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ●●● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●●● ● ●● ● ● ●● ● 10 ● ● ● ● ●● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ●● ● ●● ●● ● ● ● ● ●●●● ●●●● ●● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●●● ● ●● ● ●● ●● ● ● ●●● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ●● ● ●● ●●● ● ● ● ● ● ● ●●● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ●●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● 5 ●●● ● ●● ● ●● ● ●● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● 10 20 30 40 x Friday, 26 February 2010
  • 15. ● ● 30 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●●● 25 ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●● ● ●●● ● ● ● ● ●● ● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ●● ● ● ● ●●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ● ●● ● ● ●●●● ● ●●●● ● ● ● ● ●● ● ●● ●● ●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ●●● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ●● 20 ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●●● ● ● ● ● ● ●● ●●●● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ●● ●● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ●●● ● ● ●● ●● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ●●●● ● ● ● ● ●● ● y ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ●● ●●● ● ● ●● ● ●● ● ● ● ●● ●● ● ● ● ● 15 ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ●● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●●● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● 10 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ●●● ● ● ● ●●● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●●● ● ●●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●●● ● ● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ● ●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ●● ● ●● ● ●● ● ● ●● ● ● ●● ● ●●●● ● ● ● ● ● ● ●● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ●● ● ●● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● 5 ●● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● 10 20 30 40 x Friday, 26 February 2010
  • 16. 30 25 20 y 15 10 5 10 20 30 40 x Friday, 26 February 2010
  • 17. made missed 30 25 20 y 15 10 5 10 20 30 40 10 20 30 40 x Friday, 26 February 2010
  • 18. After spring break: Will look at three pointers and useful transformations of the data Friday, 26 February 2010
  • 19. Conditional distributions Friday, 26 February 2010
  • 20. f (x, y) X |Y = y f (x|y) = f (y) f (x, y) Y |X = x f (y|x) = f (x) Friday, 26 February 2010
  • 21. 30 25 20 15 y 10 5 0 0 10 20 30 40 50 x Friday, 26 February 2010
  • 22. Conditional [0,5] (5,10] (10,15] distribution of X 40 given Y 30 20 10 0 ..count.. (15,20] (20,25] (25,30] 40 30 20 10 0 10 20 30 40 10 20 30 40 10 20 30 40 x Friday, 26 February 2010
  • 23. [0,5] (5,10] (10,15] 40 30 20 10 0 ..count.. (15,20] (20,25] (25,30] 40 30 20 10 0 10 20 30 40 10 20 30 40 10 20 30 40 x Friday, 26 February 2010
  • 24. 30 25 20 15 y 10 5 0 0 10 20 30 40 50 x Friday, 26 February 2010
  • 25. Conditional[0,10] (10,20] (20,30] distribution of Y 60 given X 50 40 30 20 10 0 ..count.. (30,40] (40,50] 60 50 40 30 20 10 0 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 y Friday, 26 February 2010
  • 26. [0,10] (10,20] (20,30] 60 50 40 30 20 10 0 ..count.. (30,40] (40,50] 60 50 40 30 20 10 0 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 y Friday, 26 February 2010
  • 27. f (x, y) = f (y|x) f (x) f (x|y) f (y) If x and y are independent, what does that imply about f (x, y) ? Friday, 26 February 2010
  • 28. f (x, y) = f (y|x) f (x) f (x|y) f (y) If x and y are independent, what does that imply about f (x, y) ? f (x, y) = f (x) f (y) Friday, 26 February 2010
  • 29. 2d 1d Marginal n io at gr te in Joint fix in g va lu e Conditional Friday, 26 February 2010
  • 30. Why is the cdf less useful in 2d? P(X2 + Y2 < 1) P(x1 < X < x2, y1 < Y < y2) Friday, 26 February 2010
  • 31. P(x1 < X < x2, y1 < Y < y2) = (x2,y2) (x1,y1) Friday, 26 February 2010
  • 32. P(x1 < X < x2, y1 < Y < y2) = (x2,y2) F(x2, y2) (x1,y1) Friday, 26 February 2010
  • 33. P(x1 < X < x2, y1 < Y < y2) = (x2,y2) F(x2, y2) - F(x1, y2) (x1,y1) Friday, 26 February 2010
  • 34. P(x1 < X < x2, y1 < Y < y2) = (x2,y2) F(x2, y2) - F(x1, y2) - F(x2, y1) (x1,y1) Friday, 26 February 2010
  • 35. P(x1 < X < x2, y1 < Y < y2) = (x2,y2) F(x2, y2) - F(x1, y2) - F(x2, y1) (x1,y1) + F(x1, y1) Friday, 26 February 2010
  • 36. Expectation Friday, 26 February 2010
  • 37. E(aX + bY) = aE(X) + bE(Y) expectation is still a linear operator! E(XY) = E(X)E(Y) if X and Y are independent Friday, 26 February 2010
  • 38. Correlation Friday, 26 February 2010
  • 39. From: http://en.wikipedia.org/wiki/File:Correlation_examples.png Friday, 26 February 2010
  • 40. Correlation = 0.8 Friday, 26 February 2010
  • 41. Covariance Easiest to define correlation in terms of another function: covariance Cov(X, Y) = E[ (X - E(X))(Y - E(Y)) ] = σXY Cov(X, X) = Var(X) = σXX = σX2 Friday, 26 February 2010
  • 42. Alternative Is there another way to compute the covariance? (Think about the two ways of computing the variance) If X and Y are independent, what is Cov(X, Y)? Friday, 26 February 2010
  • 43. Covariance If σXY > 0, then X tends to increase when Y increases If σXY < 0, then X tends to decrease when Y increases If σXY = 0, then there is no linear relationship between X and Y (but there may be a non-linear relationship!) Friday, 26 February 2010
  • 44. Correlation σXY ρXY = σX σY σXY ρXY =√ σXX σY Y Friday, 26 February 2010
  • 45. ρ What is the range of ρ? When are X and Y most strongly positively related? When are X and Y most strongly negatively related? Friday, 26 February 2010
  • 46. Counterexample Let X be a random variable with E(X) = 0, E(X2) = 10, E(X3) = 0. Let Y = X2. Are X and Y independent? Are X and Y uncorrelated? Friday, 26 February 2010