Central Limit Theorem and Normal Distribution I have the following questions as homework where I have to decide whether they are true or false.
a) The standard deviation of the distribution of the sum of independent random variables is also called the central limit theorem.
My guess for this question is that it should be true as the variance is the sum of all the variances of the number of variables involved and thus the standard deviation is the square root of the variance.
b) When a great many n random variables are drawn from a population that is normally distributed, the distribution of the sample mean will be normal regardless of the sample size n.
For this part I don't really know how to work it out and decide if its true or false.
Can someone help me? Thank you.
 A: Strictly speaking the Central Limit Theorem states that means $\bar X_n$ of large
samples from any population, in which the variance $\sigma^2$ exists, has approximately a normal distribution for large $n,$ and that the limiting distribution as $n \rightarrow \infty$ is exactly normal.
To be specific about the approximate distribution of $\bar X_n,$ one often
says $X_n \stackrel{aprx}{\sim}\mathsf{Norm}(\mu, \sigma/\sqrt{n}),$ where $\mu$ is the population mean and $\sigma$ is the population standard deviation.
(a) It is true that $SD(\bar X_n) = \sigma/\sqrt{n},$ for all $n = 1,2,3,\dots.$ This fact
may be useful in computations for sufficiently large $n$ to obtain results about
a nearly-normal $\bar X_n,$ but it is not a limit theorem ("Central" or otherwise).
(b) The statement is true, but if one is sampling from a normal population
with mean $\mu$ and standard deviation $\sigma,$ then the exact distribution
of $\bar X_n$ is $\mathsf{Norm}(\mu, \sigma/\sqrt{n}).$ Again here, you have
a true statement that is not a limit theorem.
The CLT has to do with the behavior of $\bar X_n$ for a sample from a
non-normal distribution as $n$ becomes infinite. I believe the point of
the question is to get you to focus on the fact that the CLT is a limit theorem.
