Typo on Wikipedia's entry on Hoeffding's inequality? In Wikipedia's entry on Hoeffding's inequality, they state that if $\overline{X} := \frac 1 n \sum_{i=1}^n X_i$, then
$$ P(\overline{X}-E[\overline{X}] \ge t) \le \exp (-2n^2 t^2)$$
if we assume for simplicity that $a_i=0$ and $b_i=1$ for all $i$.
This is my first encounter with Hoeffding's inequality, so I am unfamiliar with all the different manifestations of this theorem, but regardless, all the other sources I have seen have something along the lines of $\exp (-2nt^2)$ instead. Is Wikipedia's entry a typo?
Remark: I believe that the version of the inequality where we consider $Y := \sum_{i=1}^n X_i$ instead of the average $\overline{X}$ has the bound $\exp(-2t^2)$. But, nowhere else have I seen $\exp(-2n^2 t^2)$ besides the Wikipedia entry.
Thanks!
 A: It is just a matter of how you define the random variables: the sum ($Y := \sum_{i=1}^n X_i$) or the average ($\overline{X} := \frac 1 n \sum_{i=1}^n X_i = \frac 1 n Y$), otherwise the two bounds are equivalent:
$$\Pr(Y - \mathrm{E}[Y] \geq t) \leq \exp \left( - \frac{2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right),\!$$
with $t = t'\,n$ we have:
$$\Pr(Y - \mathrm{E}[Y] \geq t'n) \leq \exp \left( - \frac{2n^2 t'^2}{\sum_{i=1}^n (b_i - a_i)^2} \right),\!$$
Note that $\Pr(Y - \mathrm{E}[Y] \geq t'n)  = \Pr(\frac{1}{n}(Y - \mathrm{E}[Y]) \geq t') = \Pr(\overline{X} - \mathrm{E}[\overline{X}] \geq t')$, so the above inequality is equivalent to 
$$\Pr(\overline{X} - \mathrm{E}[\overline{X}] \geq t') \leq \exp \left( - \frac{2n^2 t'^2}{\sum_{i=1}^n (b_i - a_i)^2} \right).$$
A: The way you copied down the equation isn't quite right. If $a_i=0$ and $b_i=1$ for all $i$, then the bound is:
$$\exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right)=\exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n 1} \right)=\exp \left( - \frac{2n^2t^2}{n} \right)=\exp \left(-2nt^2\right).$$
The form on the left is from Wikipedia, which is correct; it's the same as equation (2.6) in the source. The form on the right is what you were expecting.
