Suppose people receive a random draw which represents the probability of some event. The draws are taken randomly from a normal distribution with true mean $\mu$ and standard deviation $\sigma$.
Anyone who receives a draw less than 0 or greater than 1 discards the draw they received and obtains another draw until they sample one within the bounds of 0 and 1. The mean of people's draws must be between 0 and 1, but is necessarily closer than the true mean is to 0.5 due to truncation on both ends. Similarly the standard deviation of people’s draws is smaller than the true $\sigma$ due to truncation. Intuitively, the closer the true mean is towards 0 or 1, the greater the difference between the truncated mean and the true mean $\mu$.
Suppose we drew a small sample of points from this truncated distribution and calculate the sample mean and standard deviation.
My question is: is there some analytic solution to calculating $\mu$ using the sample set of points observed? What about $\sigma$? If there is no analytic solution, how best to approach this problem?
