17 Sampling Dist

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17 Sampling Dist

  1. 1. Stat310 Sampling distributions Hadley Wickham Monday, 22 March 2010
  2. 2. Quiz • Pick up quiz on your way in • Start at 1pm • Finish at 1:10pm • Closed book Monday, 22 March 2010
  3. 3. http://xkcd.com/715/ Monday, 22 March 2010
  4. 4. 1. Quiz 2. CLT & approximations 3. Sampling distributions 4. Example 5. More theory Monday, 22 March 2010
  5. 5. CLT Central limit theorem. The distribution of a mean is normal when gets big. Monday, 22 March 2010
  6. 6. Approximation This implies that if n is big then ... Monday, 22 March 2010
  7. 7. Sampling distributions Monday, 22 March 2010
  8. 8. Random experiment “A random experiment is an experiment, trial, or observation that can be repeated numerous times under the same conditions... It must in no way be affected by any previous outcome and cannot be predicted with certainty.” (http://cnx.org/content/m13470/latest/) i.e. it is uncertain (we don’t know ahead of time what the answer will be) and repeatable (ideally). Monday, 22 March 2010
  9. 9. Where we are Univariate random variables: an experiment with one output Bivariate random variables: an experiment with two outputs Sequences of random variables: An experiment performed repeatedly. Repeatable = i.i.d Monday, 22 March 2010
  10. 10. A sampling distribution: Summary statistics from a repeated experiment Monday, 22 March 2010
  11. 11. Definitions Sample = results of n random experiments. Random sample = result of a random experimented repeated n times. Therefore, they’re iid. Both are sequences of random variables. Statistic = A function of random variables with no unknown parameters. Monday, 22 March 2010
  12. 12. Example Spin a bottle and record the angle in degrees in which it points. Repeat. How would you write this mathematically? Monday, 22 March 2010
  13. 13. First time x1 = 205, x2 = 256, x3 = 86, x4 = 119, x5 = 16, x6 = 278, x7 = 55, x8 = 16, x9 = 295, x10 = 341, x11 = 299, x12 = 270, x13 = 118, x14 = 360, x15 = 97, x16 = 282, x17 = 42, x18 = 283, x19 = 259, x20 = 326 Monday, 22 March 2010
  14. 14. Second time x1 = 184, x2 = 344, x3 = 118, x4 = 226, x5 = 208, x6 = 106, x7 = 332, x8 = 310, x9 = 339, x10 = 95, x11 = 7, x12 = 274, x13 = 120, x14 = 346, x15 = 211, x16 = 166, x17 = 84, x18 = 102, x19 = 32, x20 = 128 Monday, 22 March 2010
  15. 15. 20 ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● 15 ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● Experiment ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●●● ● 5 ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ●● ●● ● ●● ● ● ● ● ● ● 50 100 150 200 250 300 350 Value Monday, 22 March 2010
  16. 16. 20 ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● 15 ●● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ● ● Experiment ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●●● ● 5 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ●● ●● ● ● ●● ● ● ● ● ● ● 50 100 150 200 250 300 350 Value Monday, 22 March 2010
  17. 17. 4 3 count 2 1 0 140 160 180 200 samp Monday, 22 March 2010
  18. 18. 8000 6000 count 4000 2000 0 100 150 200 250 V1 Monday, 22 March 2010
  19. 19. What will happen as I 8000 vary the number of samples I average over? (What theorem 6000 applies here?) count 4000 2000 0 100 150 200 250 V1 Monday, 22 March 2010
  20. 20. 1 2 3 400 300 200 100 0 count 4 5 400 300 200 100 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 mean Monday, 22 March 2010
  21. 21. 1 10 4000 3000 2000 1000 0 count 100 1000 4000 3000 2000 1000 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 mean Monday, 22 March 2010
  22. 22. 1 10 How can I transform 4000 this random variable to 3000 make it comparable? (What theorem applies 2000 here?) 1000 0 count 100 1000 4000 3000 2000 1000 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 mean Monday, 22 March 2010
  23. 23. 1 2 3 4 800 600 400 200 0 5 10 20 100 800 600 count 400 200 0 1000 10000 800 600 400 200 0 −400 −200 0 200 400 −400 −200 0 200 400 −400 −200 0 200 400 −400 −200 0 200 400 (mean − 180) * sqrt(n) Monday, 22 March 2010
  24. 24. 2 3 We can do the same thing 1000 for other statistics... 800 600 400 200 0 count 4 5 1000 800 600 400 200 0 0 50 100 150 200 250 0 50 100 150 200 250 sqrt(var) Monday, 22 March 2010
  25. 25. 2 3 4 600 700 500 600 500 400 500 400 300 400 300 300 200 200 200 100 100 100 0 0 0 0 50 100 150 200 250 0 50 100 150 200 0 50 100 150 5 10 20 1000 600 800 800 600 count 600 400 400 400 200 200 200 0 0 0 50 100 150 40 60 80 100 120 140 160 60 80 100 120 140 100 1000 10000 800 1000 800 800 600 600 600 400 400 400 200 200 200 0 0 0 90 95 100 105 110 115 120 98 100 102 104 106 108 110 102.503.003.504.004.505.005.5 1 1 1 1 1 1 sqrt(var) Monday, 22 March 2010
  26. 26. Theory We’ll start with the mean of normally distributed random variables, then try to extend in various ways. Monday, 22 March 2010
  27. 27. Your turn X1, X2, ... are iid N(μ, σ2) n ¯ Sn Sn = Xi Xn = n 1 Find their mgfs. What do you notice? Hint: MX (t) = exp µt + σ t 2 2 Monday, 22 March 2010
  28. 28. Reading 4.2, 4.2.1 4.2.2, 4.4 Monday, 22 March 2010

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