Question relating to Markov's inequality

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April 16th 2010, 04:08 AM
anabelle

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!
April 16th 2010, 04:48 AM
Laurent

Quote:

Originally Posted by anabelle

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.
April 16th 2010, 02:20 PM
matheagle

what about the t?
April 16th 2010, 02:35 PM
Laurent

Quote:

Originally Posted by matheagle

what about the t?

Good point. Let me rephrase:

Quote:

Originally Posted by Laurent

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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