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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen JanuaryApril 2011 Jiahua Chen () Lecture 19 JanuaryApril 2011 1 / 24 What LLNs do not answer Let X 1 , X 2 , . . . be a sequence of iid random variables with mean and variance 2 . Denote X n = n 1 ( X + 1 + X 2 + + X n ) . According to WLLN, lim n P (  X n  ) = 0 for any > 0. According to SLLN, P ( lim n X n = ) = 1. Both Laws tell us that X n . Yet how close are they? Jiahua Chen () Lecture 19 JanuaryApril 2011 2 / 24 More detailed error bound By Chebyshevs inequality, we have P (  X n  ) 2 n 2 . Letting = n 1 / 3 , it becomes P (  X n  n 1 / 3 ) 2 n 1 / 3 . Note the limit of the upper bound is still 0 as n . Jiahua Chen () Lecture 19 JanuaryApril 2011 3 / 24 More detailed error bound By Chebyshevs inequality, we have P (  X n  ) 2 n 2 . Letting = n 1 / 3 , it becomes P (  X n  n 1 / 3 ) 2 n 1 / 3 . Note the limit of the upper bound is still 0 as n . That is, the precision for X n as an estimator of is within a range of n 1 / 3 . Jiahua Chen () Lecture 19 JanuaryApril 2011 3 / 24 More detailed error bound Keep the inequality here again: P (  X n  ) 2 n 2 . Letting = n 2 / 3 , it becomes P (  X n  n 2 / 3 ) n 1 / 3 2 . Note the limit is as n . Jiahua Chen () Lecture 19 JanuaryApril 2011 4 / 24 More detailed error bound Keep the inequality here again: P (  X n  ) 2 n 2 . Letting = n 2 / 3 , it becomes P (  X n  n 2 / 3 ) n 1 / 3 2 . Note the limit is as n . The largest possible probability is 1, so the infinity upper bound is not useful. Yet it does tell us that the difference  X n  will exceed n 2 / 3 . Jiahua Chen () Lecture 19 JanuaryApril 2011 4 / 24 Tossing a die 6000 times. These analyses should reveal that n 2 / 3 <  X n  < n 1 / 3 . When n = 6000, it becomes: 0.003 <  X n  < 0.055. In general, it is easy to see that n 1 / 2 gives more tight bounds: P ( n  X n  3 ) 1 9 . That is, with 89% probability, for this die tossing example,  X n  < 0.01443376. Jiahua Chen () Lecture 19 JanuaryApril 2011 5 / 24 Tossing a die 6000 times: exact value We may directly compute P (  X n  > 0.01443376 ) = 0.0030543 = 0.3% which is much lower than the Chebyshev upper bound: 11%....
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 Spring '11
 Dr.Chen
 Probability, Variance

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