Student's T distribution was discovered by William.S. Gosset. It is a family belongs to continuous probability distributions which arises while estimating the normally distributed population mean. It is used to estimate population parameters when the population variance is unknown and the sample size is small. Student's T distribution is univariate in statistics and there are many different T distributions. The particular form of the T distribution is determined by its degrees of freedom.

Larger the degrees of freedom, closer the T distribution is to the normal density.

T Distribution is the second most commonly used probability distribution in statistics.

Given below are the important properties of t distribution.
  • T- distribution is bell-shaped and symmetric about the value zero on the horizontal axis.
  • Shape of the curve approaches the standard deviation as the sample size n increases.
  • Area in the tails of the T-distribution is larger than the area in the tails of the normal distribution.
  • Mean is zero and the random variable 't' can have any value between -$\infty$ and $\infty$. Most of the probability density is found in the vicinity of t = 0.

T Distribution
A random sample of size n is drawn from a normal population with standard deviation $\sigma$ and mean $\mu$. Let $\bar{x}$ denote sample mean and s be the standard deviation. Then, we have
t = $\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}$
This has a T distribution with n - 1 degrees of freedom. The T distribution is also known as the "Student's T Distribution".
Where,
$\mu$: Mean of the population.
$\bar{x}$: Mean of the sample.
Given below is the t-distribution table:

T Distribution Table
In T distribution, critical values are calculated according to the degrees of freedom and the probabilities of two alpha values. In T distribution, the spread is more than that of the standard normal distribution.
When the degrees of freedom exceed 30 critical values that are not tabulated for every case, it is preferred to round up the critical value to the nearest degrees of freedom.
Multivariate T distribution is a generalization of the univariate Student's T to two or more variables. It plays an important role in the application of the simultaneous test procedures and ranking and selection procedures.

The probability density function for multivariate probability t distribution with m degrees of freedom is given below:

g(t) =  $\frac{1}{\sqrt{\nu }\beta \left ( \frac{1}{2},\frac{\nu }{2} \right )}$$\left ( 1+\frac{t^{2}}{m} \right )$$^{-(\frac{m+1}{2})}$
Non central T distribution takes on a wide range of shapes, with the degree of asymmetry varying with the degrees of freedom which depends on the distance. It approaches the normal distribution in shape, as n increases, but it does so very slowly.

The non central distribution f(m) is illustrated as below:

f(m) = $\frac{\sqrt{k}(\bar y - m)}{s}$

The value of sample deviation $s^{2}$ is $\frac{1}{k - 1}$$\sum_{i = 1}^k (y_i - \bar y)^2$.
The t distributions used in the simulations are conventional symmetric generalized t distributions with 1 standard deviation and 0 mean.

Its pdf is as follows:

p(x) = C$^{-1}_{v,\lambda}$ [1 $\pm$ $\frac{x}{(v + x^2)}^{\frac{1}{2}}]$$^{\frac{v\pm\lambda+1}{2}}$

where,
 
C$^{-1}_{v,\lambda}$ = 2$^{v-1}B(\frac{v+\lambda}{2}, \frac{v-\lambda}{2})$ and B is a beta distribution.

v = Degrees of freedom
$\lambda$ = Possible skewness