Share of success in a binomial process as N goes to infinity

I have a binomial process with probability of success p. Let k be number of success in when the number of trails is N. I have defined s as the random variable of the share of success $\ s=\frac{k}{N}$

i need to find the limiting distribution of s as $\ \lim_{N\to \infty} s$

I have tried to go about it as follows. The chance of exactly k success is $C_k^N *p^k*(1-p)^(N-k)$

Then m=[N*s] where [] is the least integer function.The limit distribution will be the limiting value of $\ \lim_{N\to \infty}C_m^N *p^k*(1-p)^\left(N-m\right)$ I tried to draw some inferences using simulation.It seems that the limiting distribution of the share of success is a normal with mean p empirical distribution of share of success

• The central limit theorem shows that the distribution actually is asymptotically normal Commented Sep 9, 2017 at 19:24
• You might be interested in reading about the De Moivre- Laplace theorem. Commented Sep 9, 2017 at 19:26
• Hi Peter,what is the mean and variance of that distribtion? Just by eyeballing the distribution,it seems the mean is also p. Commented Sep 9, 2017 at 19:48
• @kangkanDc Yes, this makes sense. In the usual binomial distribution, the variance would be $np(1-p)$, but I am not sure whether we can use this result for this concrete distribution. Unfortunately, I have no idea what the variance is. Commented Sep 9, 2017 at 21:08