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Chapter 8: Q. 8.10 (page 393)

It Xis a Poisson random variable with a mean位, showing that fori<位,

P{X鈮�i}鈮�e-位(e位)iii

Short Answer

Expert verified

Therefore,

ddt位et-1-it=0鈬�et=i位

Hence proved.

Step by step solution

01

Given Information.

Xis a Poisson random variable with a mean位,

02

Explanation.

We are going to be using Chernoff bounds. Specifically, it Xis a random variable withMGFM(t)andi>0,t<0

鈬�P(X鈮�i)鈮�e-itM(t)

03

Explanation.

Xpoiss(位)then the MGFisM(t)=e位et-1. Then using Chernoff bound:

i<位,P(X鈮�i)鈮�e-ite位et-1=e位et-1-it

We want tto minimize this. Since exis a strictly increasing function, the minimizing argument is given by the first-order conditions of

位et-1-it

So we derive:

ddt位et-1-it=0鈬�et=i位

Sincei鈮�位this means that t<0and so the Chernoff bound is valid.

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