Question

For a large number of independent draws from a box, will standard deviation, range of draws and proportion of 1's in draws from a 0-1 box have a normally distributed probability histogram?

Answer 1

Yes for All

From a 0-1 box in which the 0's and 1's are kept in a certain proportion, the event of certain number of independent draws is a Bernoulli trial having Binomial distribution. By independent draws, we essentially refer to replacing back the drawn ball, so that the probability of 0-1 does not change in next draw. This probability distribution of X, the number of 1's drawn in n draws is given by

When n becomes large, then Central Limit Theorem implies that the Binomial distribution can be approximated to the Normal distribution. Since each draw denotes an independent, identically distributed random variable, hence their sum will approach the Normal distribution for reasonably large values of N. The parameters, mean and variance are obtained as

That is, the mean of normal distribution is µ = np, and its variance is σ2 = np(1-p)