Random Samples
A random sample is a collection of independent random variables X_{1}, X_{2},..., X_{n}, all with the same probability distribution.
The Sample Mean
The sample mean,
of a random sample X_{1}, ... X_{n} is given by:
If the random varables each have a normal distribution (with mean m and variance s^{2}), then the sample mean has a normal distribution with mean m and variance s^{2}/n, in other words:
Expectation and Variance
is itself a random variable and so has an expectation and variance.
These are easy to calculate, for example:
E(
) = E ( SX_{i}/n ) = (1/n)E( X_{1} + X_{2} + ... + X_{n}) = (1/n)[E(X_{1}) + E(X_{2}) + ... + E(X_{n})]
If each of the random variables X_{1}, ... X_{n }have expectation m, then this is equal to (1/n)(n m) = m .

E(
Image) = m
Similarly, if each of the random variables X_{1}, ... X_{n }has variance s^{2}, it can be shown that:

Var(
Image) = (s^{2})/n
The Sample Variance
The sample variance is given by: