Estimate the normal distribution of the mean of a normal distribution given a set of samples?
The same can be done through any of the statistical estimation technique such as maximum likelihood or Minimum variance unbiased estimator(MVUE). Generally maximum likelihood is the easiest way to evaluate the mean and variance of the distribution. If you perform Maximum likelihood method the estimated mean of the normal distribution is the sample mean. Given x1,x2.....xn are the samples 

Estimated mean u = (x1 + x2 + ... + xn)/n
Similarly the estimated variance is  Σ(xi - u)^2/n

Both of these are obtained by setting the partial derivative of the likelihood function with respect to the mean and the standard deviation to zero.

Hope this helps.