probability - What is the analytical expression which shows the convergence of a 6 sided fair dice's expected value to 3.5 as a function of the number of rolls(N)?

What is the analytical expression which shows the convergence of a 6 sided fair dice's expected value to 3.5 as a function of the number of rolls(N)? I realize that there may be a need for a confidence interval.

Here is how to apply the central limit theorem as an approximation for large N rolls. The distribution of the sample mean multiplied by N−−√ is, thanks to the Central Limit Theorem, approximately normal with mean of the population μ and standard deviation σ. So the distribution of the mean (or the sum) is approximately normal for large N with mean μ (or Nμ ) and standard deviation σ/N−−√ (or σN−−√) .

The Central limit theorem can be demonstrated with characteristic functions. That for a discrete uniform distribution such as a 6-sided die is at en.wikipedia.org/wiki/Discrete_uniform_distribution.

Any help is greatly appreciated.

Answer

The Central Limit Theorem is an asymptotic result. I am not sure that it is what you are looking for. I imagine you are looking for some finite sample result. I will present one possible way of doing it.

Let $X_i$ be a random variable taking values in $\{1,2,3,4,5,6\}$, which denotes the output of the the $i^{th}$ trial. Let $S_n = \sum_{i=1}^nX_i$, then by applying Chebyshev's inequality (and noting that $var(S_n)=n \times var(X_1)$, since $(X_i)_{i=1}^n$ are i.i.d.), we have

$P(|\frac{S_n}{n}-\mu| > \epsilon) \leq \frac{1}{n\epsilon^2}var(X_1)$

Thus for large value of $n$, this probability goes to zero for any $\epsilon >0$. It is possible to get much tighter results than this by using better concentration inequalities.