# Question relating to Markov's inequality

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• Apr 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

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

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 $\displaystyle M'(0)$, but I'm not sure how to do this without any more information about X and its mgf.

Any help would be appreciated!
• Apr 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

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

where M(t) is the moment generating function of X. I think that Markov's inequality would be useful here

Notice that $\displaystyle X\geq a$ if, and only if $\displaystyle e^X\geq e^a$. Thus, $\displaystyle P(X\geq a)=P(e^X\geq e^a)$ and apply Markov's inequality now. By the way, no non-negativity is needed here.
• Apr 16th 2010, 02:20 PM
matheagle
what about the t?
• Apr 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 $\displaystyle t>0$. Notice that $\displaystyle X\geq a$ if, and only if $\displaystyle e^{tX}\geq e^{ta}$. Thus, $\displaystyle P(X\geq a)=P(e^{tX}\geq e^{ta})$ and apply Markov's inequality now. By the way, no non-negativity is needed here, except for $\displaystyle t$.