Will the distribution of the sample means follow the uniform distribution?

Answer 1

The distribution of the sample means will, as the sample size increases, follow the normal distribution. This is true for any given distribution (e.g. does not need to be a normal distribution). This concept is from the central limit theorem. It is one of the most important concepts in statistics, along with the law of large numbers. An applet to help you understand this concept is located at: http:/www.stat.sc.edu/~west/javahtml/CLT.html