Prove an extension of the Chernoff bound .Let X=i=1nXi , where the Xi 's are independent 0-1 random variables.Let =E[X] .Choose any L and H such that LH .Prove the following:

1) For any >0 ,Pr(X(1+)H)((1+)(1+)e)H.

For any 0<<1 , Pr(X(1)L)((1)(1)e)L.

2) An (,) -confidence interval for an unknown constant T is a (random) interval [A,A+] such that Pr(T[A,A+]).

Let X1,...,Xn be n independent observations such that Xi=1 with unknown probability P , otherwise Xi=0.

Let X=n1i=1nXi . Use the question 1) to bound the error probability of theconfidence interval [X,X+] for P.