The t-distribution was discovered by W. S. Gosset and R.A. Fisher. The entries in Student’s *t* table entries are the critical values (percentiles) for the *t* distribution. The applications of Student’s t distribution are related to (i) the sampling distribution of the mean $\overline{x}$, (ii) the distribution of a difference $(\overline{x}_1 – \overline{x}_2)$ of two independent populations, (iii) the distribution of two paired (dependent) populations, and (iv) the significance of correlation coefficient. It is also used for constructing confidence intervals for small samples. The Student’s t distribution is a crucial tool in statistical analysis, especially when dealing with small sample sizes. It helps us make informed decisions based on our data, even when the population standard deviation is unknown.

The Student’s *t* variable can be generated by dividing the standard normal random variable ($Z$) with the square root of a $\chi^2_{v}$ random variable. The $\chi^2_v$ is itself divided by its parameter $v$. That is

\begin{align*}

t_v &= \frac{x – \mu }{s_v} = \frac{\tfrac{(x-\mu)}{\sigma} }{\sqrt{\dfrac{\frac{v\times s^2_v}{\sigma^2} } {v}}}\\

&= \frac{Z}{\sqrt{\dfrac{\chi^2_v}{v}}}

\end{align*}

where

- $Z$ is from a standard normal distribution
- $chi _{v}^{2}$ is a Chi-square random variable with $v$ degrees of freedom
- $Z$ and $chi _{v}^{2}$ are independent of each other

### PDF of Student’s t Distribution

The PDF of *t* having $v$ degrees of freedom is

$$p(t_v) = K_v (1+\frac{t^2}{v})^{-\frac{v+1}{2}}$$

where

$$K_v = \frac{\Gamma \left[ \frac{(v+1}{2} \right]}{\sqrt{v\pi} \left(\frac{v}{2}\right) }$$

The *t* distribution is symmetric about zero and wider than normal density. It has one mode and it tends to be normal as $v\rightarrow \infty$. Note that $\Gamma(x)$ indicates the Gamma function.

### Moments of t Distribution

Since the *t* distribution is symmetric and its PDF is centered at zero, the expectation (average), the median, and the mode are all zero for the t distribution with $v$ degrees of freedom. The variance ($\sigma^2$) equals $\frac{v}{v-2}$ and kurtosis is $\frac{6}{v-4}$.

For bivariate normal population, the distribution of correlation coefficient $r$ is linked with Student’s *t* distribution through transformation:

$$\frac{r}{\sqrt{\frac{1-r^2}{n-2}}}\rightarrow t_{n-2}$$

### Generation of Pseudo Random t Variates

The following algorithm can be used to generate random variates from Student’s $t(v)$ distribution using serially generated independent uniform $U(0,1)$ random variates. For example,

Let $n=v$ (the degrees of freedom)

$C = -2n$**Repeat**

$t = 2 \times U(0, 1) – 1$

$u = 2 \times U(0, 1) – 1$

$r = t^2 + u^2$**Until** $r < 1$

**Return**

$t \times \sqrt{\frac{n \times (r^C – 1)}{r}}$

### Student’s t Table

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