Apply the concept of Central Limit Theory If random variable

Apply the concept of Central Limit Theory.

If random variables X1, X2, X3, ... Xn follows Normal(0,1) distribution, the mean (of X1, X2, X3, ... Xn) will be normally distributed.

True or False?

Solution

TRUE.

This is true even when the underlying distribution is not normal. How much more if they are really normally distributed.

Actually, the central limit theorem REQUIRES that the sample size is large enough, so that the sample means become more normally distributed. But in this case, the underlying distribution is normal already.