Student's distribution

If is a sample of size N drawn from a normal distribution with mean E(X) and variance , then

and

are estimators of the mean E(X) and of the variance of the estimator . The quotient

is described by Student's distribution (also called the t-distribution) with f=N-1 degrees of freedom. Its probability density is

where denotes Euler's gamma function. The t-distribution has the properties

For File:Hepa img1068.gif it approaches the standard normal distribution.