The t distribution is another that is closely associated with random samples from a normal distribution.

Definition

If \(Z \sim \NormalDistn(0, 1)\) and \(Y \sim \ChiSqrDistn(k \text{ df})\) are independent random variables, then

\[ T \;\;=\;\; \frac{Z}{\sqrt{\diagfrac{Y}{k}}} \]

has a distribution called a t distribution with k degrees of freedom,

\[ T \;\;\sim\;\; \TDistn(k \text{ df}) \]

We cannot derive the pdf of the t distribution here, but simply state it without proof.

Probability density function of t distribution

If a random variable has a t distribution with \(k\) degrees of freedom then its probability density function is

\[ f(x) \;\;=\;\; \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2}) \sqrt{k\pi}} \left(1 + \frac{x^2}{k} \right)^{-\frac {k+1}{2}} \]

Probabilities and quantiles

There is no explicit formula for the t distribution's cumulative distribution function, but its cumulative probabilities can be evaluated numerically by most statistical computer programs. In Excel, the probability of a t distribution with \(\langle df \rangle\) degrees of freedom being less than a value \(\langle y \rangle\), \(P(Y \lt \langle y \rangle)\) can be found by typing the following into an Excel spreadsheet cell.

=T.DIST(\(\langle y \rangle\), \(\langle df \rangle\), TRUE)

In a similar way, the \(\langle p \rangle\)'th quantile of the distribution can be found with

=T.INV(\(\langle p \rangle\), \(\langle df \rangle\))