# Statistics and Their Distributions

Posted by Beetle B. on Wed 07 June 2017

When we take a sample and calculate its mean and standard deviation, this is treated as a random variable for the population mean/standard deviation.

We treat each observation as a random variable.

The random variables $$X_{1},\dots,X_{n}$$ form a random sample of size $$n$$ if:

1. The $$X_{i}$$ are independent.
2. Every $$X_{i}$$ has the same probability distribution.

Note that the second is not true when sampling without replacement. We treat it as a good approximation, though, if $$n/N\le0.05$$

The above two conditions are the same as saying the $$X_{i}$$ are independent and identically distributed (iid).

## Normal Distribution

For a normal distribution, the expected value of the sample mean is:

\begin{equation*} E(S)=\sqrt{\frac{2}{n-1}}\Gamma\left(\frac{n}{2}\right)\frac{\sigma}{\Gamma\left(\frac{1}{2}(n-1)\right)} \end{equation*}