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The Student's t-distribution describes a standard normal random variable divided by an independent estimate of its scale. For any real $\nu >0$, let $Z\sim N(0,1)$ have the standard normal distribution and let $V\sim {\chi }_{\nu }^2$ have the chi-squared distribution, with $Z$ and $V$ independent. ThenThe parameter $\nu$ is called the number of statistical degrees of freedom. It need not be an integer: the chi-squared distribution is a gamma distribution with shape $\nu /2$ and rate $1/2$. The independence assumption is essential; specifying only the two marginal distributions does not determine the probability distribution of their ratio.

Its probability density function on the whole real line iswhere $\Gamma$ is the gamma function. Every member is a symmetric distribution about zero. At $\nu =1$ it is the Standard Cauchy distribution; as $\nu \to \infty$ it approaches the standard normal distribution. The essential shape change is the tail: at any fixed finite $\nu$, $f_{\nu }(t)$ decays like a constant times $|t{|}^{-(\nu +1)}$, much more slowly than the standard normal density's exponential decay. Thus it is a heavy-tailed distribution. ^[ref]

The left panel shows the centers; the right panel uses a logarithmic scale to reveal the tails that a linear plot hides. All four curves have location zero and scale one. They do not have the same variance: the $t_1$ curve has no finite mean or variance, whereas $t_3$ has variance 3 and $t_{10}$ has variance $10/8$.

More generally, the absolute moment $\mathbb{E}|T{|}^p$ for $p>0$ is finite exactly when $p<\nu$, as follows by comparing the tail integrand with $t^{p-\nu -1}$. ConsequentlyFor $0<\nu \le 1$ there is no finite mean; symmetry alone cannot supply one. For $1<\nu \le 2$ the mean exists but the variance is infinite. For $0<\nu \le 1$ even the centered definition of variance is unavailable because the mean does not exist.

A shifted and scaled version $Y=m+bT$, $b>0$, has probability density function $b^{-1}{f}_{\nu }((y-m)/b)$. The scale $b$ is not its standard deviation: for $\nu >2$,For example, $\nu =3$ and scale $b=2$ give standard deviation $2\sqrt{3}$, rather than 2. The distinction matters when using a Student's t-distribution to model data with occasional extreme values, as well as when interpreting a Student t confidence interval.

For an independent statistical sample from a gamma distribution with known shape $\lambda >0$ and unknown rate $\theta >0$, test $\theta ={\theta }_0$ against $\theta >{\theta }_0$. The alternative-to-null likelihood ratio for any larger rate ${\theta }_1$ is $({\theta }_1/{\theta }_0{)}^{n\lambda }\exp \{-({\theta }_1-{\theta }_0)S\}$, where $S={\sum }_i{X}_i$. This decreases in $S$, so the Neyman-Pearson lemma gives one common lower-tail ordering for all alternatives.

Let $G_a$ be the unit-rate gamma distribution cumulative distribution function with shape $a$. The uniformly most powerful test of exact test size $\alpha \in (0,1)$ rejects when $S<c_{\alpha }$, where $c_{\alpha }=G_{n\lambda }^{-1}(\alpha )/{\theta }_0$. Its power function is displayed above. If $n\lambda =1$, the sum has an exponential distribution, so $c_{\alpha }=-\log (1-\alpha )/{\theta }_0$ and $\pi (\theta )=1-(1-\alpha {)}^{\theta /{\theta }_0}$. Monotonicity of $G_a(\theta c_{\alpha })$ also makes this test valid and uniformly most powerful for the larger null hypothesis $\theta \le {\theta }_0$.

For independent samples from an exponential distribution, a test of rate at most ${\lambda }_0$ against rate greater than ${\lambda }_0$ rejects for small sample sums. All simple-alternative likelihood ratios against ${\lambda }_0$ decrease with that sum; its gamma distribution calibrates the size at ${\lambda }_0$. Monotonicity of the rejection probability in the rate proves that this is a uniformly most powerful test for the composite hypotheses.