📂Probability Distribution

t-Distribution

Definition ¹

A continuous probability distribution $t \left( \nu \right)$, known as the t-distribution, is defined for degrees of freedom $\nu > 0$ as having the following probability density function: $$ f(x) = {{ \Gamma \left( {{ \nu + 1 } \over { 2 }} \right) } \over { \sqrt{\nu \pi} \Gamma \left( {{ \nu } \over { 2 }} \right) }} \left( 1 + {{ x^{2} } \over { \nu }} \right)^{- {{ \nu + 1 } \over { 2 }}} \qquad ,x \in \mathbb{R} $$

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• $\Gamma (\nu)$ is the gamma function.

Description

The t-distribution is widely known for its discovery and publication by William S. Gosset, who worked at the Guinness brewery, still famous for its beer. At that time, being bound to the company, he submitted his work under the pseudonym Student, hence it is also called the Student’s t-distribution. For freshmen in statistics, it is initially encountered for small samples, which are assumed to follow a normal distribution but in reality do not exceed 30 samples. It is considered to converge to the normal distribution when $\nu \ge 30$.

Meanwhile, the distribution when $\nu = 1$ is called the Cauchy distribution.

Basic Properties

Moment Generating Function

• [1]: The moment generating function does not exist for the $t$-distribution.

Mean and Variance

• [2]: If $X \sim t (\nu)$ then $$ \begin{align*} E(X) =& 0 & \qquad , \nu >1 \\ \operatorname{Var}(X) =& {{ \nu } \over { \nu - 2 }} & \qquad , \nu > 2 \end{align*} $$

Theorem

Let us assume two random variables $W,V$ are independent and $W \sim N(0,1)$, $V \sim \chi^{2} (r)$.

$k$th Moment

• [a]: If $k < r$, then $\displaystyle T := { {W} \over {\sqrt{V/r} } }$ is the $k$th moment and $$ E T^{k} = E W^{k} {{ 2^{-k/2} \Gamma \left( {{ r } \over { 2 }} - {{ k } \over { 2 }} \right) } \over { \Gamma \left( {{ r } \over { 2 }} \right) r^{-k/2} }} $$

Derived from Standard Normal and Chi-square Distributions

• [b]: $${ {W} \over {\sqrt{V/r} } } \sim t(r)$$

Deriving Standard Normal Distribution as the Limiting Distribution of Student’s t-Distribution

• [c]: If $T_n \sim t(n)$ then $$ T_n \ \overset{D}{\to} N(0,1) $$

Deriving the F-Distribution

• [d]: A random variable $X \sim t(\nu)$ following a t-distribution with degrees of freedom $\nu > 0$ is defined as $Y$, and follows an F-distribution $F (1,\nu)$. $$ Y := X^{2} \sim F (1,\nu) $$

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• $N \left( \mu , \sigma^{2} \right)$ is a normal distribution with mean $\mu$ and variance $\sigma^{2}$.
• $\chi^{2} \left( r \right)$ is a chi-square distribution with degrees of freedom $r$.

Proof

[1]

The existence of the moment generating function for a random variable implies the existence of all $k$th moments for every $k \in \mathbb{N}$. However, as theorem [a] states, the $k$th moment of the t-distribution exists only when $k < r$, thus the moment generating function cannot exist.

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[2]

Using the moment formula [a].

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[a]

Chi-square distribution moments: Let $X \sim \chi^{2} (r)$. If $k > - r/ 2$, then the $k$th moment exists and $$ E X^{k} = {{ 2^{k} \Gamma (r/2 + k) } \over { \Gamma (r/2) }} $$

Multiplying both sides of $k < r$ by $-1/2$ results in $-k/2 > -r/2$, hence $$ \begin{align*} E T^{k} =& E \left[ W^{k} \left( {{ V } \over { r }} \right)^{-k/2} \right] \\ =& E W^{k} E \left( {{ V } \over { r }} \right)^{-k/2} \\ =& E W^{k} {{ 2^{-k/2} \Gamma \left( {{ r } \over { 2 }} - {{ k } \over { 2 }} \right) } \over { \Gamma \left( {{ r } \over { 2 }} \right) r^{-k/2} }} \end{align*} $$

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[b]

Derived directly from the joint density function.

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[c]

Using the Stirling approximation on the probability density function.

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[d]

Circumventing by the ratio of chi-square distributions.

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Code

Below is Julia code displaying the probability density functions for the Cauchy distribution, t-distribution, and Cauchy distribution.

@time using LaTeXStrings
@time using Distributions
@time using Plots

cd(@__DIR__)

x = -4:0.1:4
plot(x, pdf.(Cauchy(), x),
 color = :red,
 label = "Cauchy", size = (400,300))
plot!(x, pdf.(TDist(3), x),
 color = :orange,
 label = "t(3)", size = (400,300))
plot!(x, pdf.(TDist(30), x),
 color = :black, linestyle = :dash,
 label = "t(30)", size = (400,300))
plot!(x, pdf.(Normal(), x),
 color = :black,
 label = "Standard Normal", size = (400,300))

xlims!(-4,5); ylims!(0,0.5); title!(L"\mathrm{pdf\,of\, t}(\nu)")
png("pdf")