# Student-t distribution¶

## Story¶

The story of the Student-t distribution largely derives from its relationships with other distributions. One way to think about it is as like the Normal distribution with heavier tails.

## Parameters¶

The Student-t distribution is symmetrically peaked, and its peak is located at $$\mu$$, the location paramter. The peak’s width is dictated by scale parameter $$\sigma$$, which is positive. Finally, the shape parameter, called “degrees of freedom,” is $$\nu$$. This last parameter imparts the distribution with heavy tails for small $$\nu$$.

## Support¶

The Student-t distribution is supported on the set of real numbers.

## Probability density function¶

\begin{align} f(y;\nu, \mu, \sigma) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\pi \nu \sigma^2}}\left(1 + \frac{(y-\mu)^2}{\nu \sigma^2}\right)^{-\frac{\nu + 1}{2}}. \end{align}

## Moments¶

Mean: $$\mu$$ for $$\nu > 1$$, otherwise undefined.

Variance: $$\displaystyle{\frac{\nu}{\nu - 2}}$$ for $$\nu > 2$$. If $$1 < \nu < 2$$, then the variance is infinite. If $$\nu \le 1$$, the variance is undefined.

## Usage¶

Package

Syntax

NumPy

mu + sigma * rg.standard_t(nu)

SciPy

scipy.stats.t(nu, mu, sigma)

Stan

student_t(nu, mu, sigma)

## Notes¶

• Only the standard Student-t distribution ($$\mu = 0$$ and $$\sigma = 1$$) is available in NumPy. You can still draw out of the Student-t distribution by performing a transformation on the samples out of the standard Student-t distribution, as shown in the usage, above.