In a histogram, we can write $f\{x_0\}$ as the histogram frequency for the bar $x=x_0$, and as it comes from a distribution, it should have a distribution too, which I don't know how to get.
If I roll a dice n times, we will have an histogram close to a flat $1/6$ frequency. If I roll it again and again, that frequency will approach that figure with some distribution with a known mean of $1/6$ and a deviation equal to ... (??).
The Discrete Uniform is the simplest case.
If I have a Discrete Distribution, i.e. a perfect dice $x\sim U(p)$ (uniform with $p$ values from $x=0$ to $x=p-1$), how should I obtain the Discrete Distribution of the frequency for a given value $x=x^0$ for a given number of realizations $x_k,k=1...n$? $$ f\{x_0\}=\frac1n \sum_{\ \ k=1 \\ x_k=x^0}^{n}x_k $$
For example, if $n=1$, $f\sim U(2)$, but with i.e. $p=2,n=2$ it is not uniform, but some triangular discrete distribution (??) having $E\{f\}=1/p$.
I know that the Expected Value of $f\{x_0\}$ must be $E\{f\{x_0\}\}=P\{x=x_0\}=\frac1p$, but I don't know how to get its standard deviation and distribution.