Thread: Problem 1.9
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Old 05-12-2016, 03:18 AM
waleed waleed is offline
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Join Date: May 2016
Posts: 5
Default Re: Problem 1.9

Originally Posted by MaciekLeks View Post
How do you know that \prod_{n=1}^N P[u_n \geq \alpha] \leq (e^{-s\alpha}U(s))^N? I think that is a problem in your proof that you assumed that the joint probability works with Problem 1.9(b) inequality.

To proof (b) I went this way:

1. I used Markov Inequality \mathbb{P}[u\geq\alpha]\leq\frac{\mathbb{E}_{u}[u]}{\alpha}

2. Problem 1.9(a) gave me this: \mathbb{P}[t\geq\alpha]=\mathbb{P}[e^{sNt}\geq e^{sN\alpha}]\leq\frac{\mathbb{E}[e^{sNt}]}{e^{sN\alpha}}, hence \mathbb{P}[u\geq\alpha]\leq\frac{\mathbb{E}_{u}[e^{sNu}]}{e^{sN\alpha}}

Using this the rest of the proof is quite nice to carry out.
I don't think the condition right
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