ECE109 Disc10.pdf - UC San Diego J Connelly ECE 109...

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UC San Diego J. Connelly ECE 109 Discussion 10 Notes Problem 10.1 Suppose the PDF of X is f X ( u ) = braceleftBigg 1 486 u 2 if 9 < u 9 0 otherwise. and let Q ( δ ) = P ( | X E [ X ] | > δ ) . (a) Determine Q ( δ ) in terms of δ > 0 . (b) Use the Chebyshev inequality to obtain an upper bound on Q ( δ ) . (c) Evaluate Q ( δ ) and the Chebyshev bound for δ = 1 , 2 , 3 , . . . , 8 , and 8 . 5 , 8 . 9 . Solutions (a) We have E [ X ] = integraldisplay 9 - 9 1 486 u 3 du = 0 and so Q ( δ ) = P ( | X | > δ ) = P ( X < δ ) + P ( X > δ ) . Since X [ 9 , 9] , if δ > 9 , then P ( X < δ ) = P ( X > δ ) = 0 , and if δ [0 , 1] , we have P ( X < δ ) = integraldisplay - δ - 9 1 486 u 2 du = 1 2 δ 3 1458 and P ( X > δ ) = integraldisplay 9 δ 1 486 u 2 du = 1 2 δ 3 1458 . Thus Q ( δ ) = P ( | X E [ X ] | > δ ) = braceleftBigg 1 δ 3 729 if δ [0 , 9] 0 if δ > 1 . (b) We have V ar [ X ] = E [ X 2 ] E [ X ] = E [ X 2 ] = integraldisplay 9 - 9 1 486 u 4 du = 243 5 . Chebyshev’s inequality tells us Q ( δ ) = P ( | X E [ X ] | > δ ) V ar [ X ] δ 2 = 243 5 δ - 2 Please report any typos/errors to [email protected]
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(c) We have δ Q ( δ ) Chebyshev Bound 1 0 . 999 48 . 6 2 0 . 989 12 . 15 3 0 . 963 5 . 4 4 0 . 912 3 . 04 5 0 . 829 1 . 94 6 0 . 704 1 . 35 7 0 . 529 0 . 992 8 0 . 298 0 . 760 8 . 5 0 . 158 0 . 672 8 . 9 0 . 033 0 . 614 In this case (and, in fact, in most cases), Chebyshev’s inequality does not provide a very tight upper bound. However, as we have seen, Chebyshev’s inequality is crucial for proving the law of large numbers, which is one of the single most important results in probability theory. In
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