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Math Help - Bernstein's inequality (please help)

  1. #1
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    Bernstein's inequality (please help)

    A) Prove Bernstein's inequality: If X is a random variable for which Mx(t) exists for some t>0, the P(X>=x) <= Mx(t)e^(-tx). Assuming X is continuous.

    B) If X~N(0,1), then Mx(t)=e^((t^2)/2) for all t.
    Thus, by A), P(X>=x)<=e^((t^2)/2 - tx).
    For x=2, find the value of t for which this inequality is sharpest.
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  2. #2
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    I'm assuming M_x(t) is the moment generating function. You might want to write out the expression for the moment generating function explicitly in integral form. Similarly, express P(X\geq x) as an integral. Compare the two and play around with it to get the desired inequality.
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  3. #3
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    Quote Originally Posted by ninano1205 View Post
    A) Prove Bernstein's inequality: If X is a random variable for which Mx(t) exists for some t>0, the P(X>=x) <= Mx(t)e^(-tx). Assuming X is continuous.

    B) If X~N(0,1), then Mx(t)=e^((t^2)/2) for all t.
    Thus, by A), P(X>=x)<=e^((t^2)/2 - tx).
    For x=2, find the value of t for which this inequality is sharpest.
    For A), you can write, for t>0, P(X\geq x)=P(e^{tX}\geq e^{tx}) and apply Markov inequality.
    In B), you are asked for the t for which the given right-hand side is minimum, so you have to study the variations of the function t\mapsto e^{(t^2)/2 - tx} (differentiate, study the sign of the derivative,...) to find which value of t minimizes it.
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