I have independent random values $x_1 < ... < x_n$ with the same distribution function. I want to find expected value of this distribution by my . Of course it will be $X = \frac{1}{n} \sum_i x_i$
Probability $P(|X - E(x)| < \epsilon) > 1- \delta$
The question is how to find the number $n$ to be accurate about $\epsilon$ and $\delta$ (they equal 0.01)? I want to find FULL proof for that accuracy. And I don't want to calculate dispersion, because it will cause same difficulties.
I had 2 variants now - Chebyshev Inequality and Hoeffding's Inequality.
First is to complicated - it needs $n> 1000000$
Second needs a precise interval for $x$ . I can estimate it theoretically, but It becomes too big, much bigger than on practice. Maybe there is some variant of HI where bounds for $x$ are changed to $x_1$ and $x_n$ ?
I also looked at Central limit theorem, but I didn't find the proof for accuracy for my $X$.
Thanks, Vasily.