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Math Help - Question relating to Markov's inequality

  1. #1
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    Question relating to Markov's inequality

    I've been given a question as follows: if Y is a non-negative random variable, I want to show that

    P(X\geq a)\leq e^{-ta}M(t),

    where M(t) is the moment generating function of X. I think that Markov's inequality would be useful here, and I know that I can express E(X) as M'(0), but I'm not sure how to do this without any more information about X and its mgf.

    Any help would be appreciated!
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  2. #2
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    Quote Originally Posted by anabelle View Post
    I've been given a question as follows: if Y is a non-negative random variable, I want to show that

    P(X\geq a)\leq e^{-ta}M(t),

    where M(t) is the moment generating function of X. I think that Markov's inequality would be useful here
    Notice that X\geq a if, and only if e^X\geq e^a. Thus, P(X\geq a)=P(e^X\geq e^a) and apply Markov's inequality now. By the way, no non-negativity is needed here.
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  3. #3
    MHF Contributor matheagle's Avatar
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    what about the t?
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  4. #4
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    Quote Originally Posted by matheagle View Post
    what about the t?
    Good point. Let me rephrase:

    Quote Originally Posted by Laurent View Post
    Let t>0. Notice that X\geq a if, and only if e^{tX}\geq e^{ta}. Thus, P(X\geq a)=P(e^{tX}\geq e^{ta}) and apply Markov's inequality now. By the way, no non-negativity is needed here, except for t.
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