# Accuracy of empirical cumulative distribution function

I have a random variable with an unknown distribution and I want to find its cumulative distribution function. I sample the distribution $N$ times, with $$X_1, \dots, X_N$$ being iid random variables. I then plot the empirical cumulative distribution function.

Am I correct in assuming that if I let $N$ be such that $$N \geq \frac{1}{2\epsilon^2}\ln(\frac{2}{\alpha})$$ Then my empirical distribution function is within $\epsilon$ of the actual cumulative distribution function at all points with a $1-\alpha$ confidence?

Yes, if $$N \geq \frac{1}{2\epsilon^2}\ln(\frac{2}{\alpha})$$, then $$\mathbb{P}\left(\sup_{x \in \mathbb{R}}|F_N(x) - F(x)| > \epsilon \right) \le 2 e^{-2N\epsilon^2} \le \alpha,$$ which is what you want.