Student's t-Distribution with One Degree of Freedom is Standard Cauchy Distribution

From ProofWiki
Jump to navigation Jump to search

Theorem

The Student's $t$-distribution with one degree of freedom is a special case of a standard Cauchy distribution.


Proof

Let $X$ be a continuous random variables with a Student's $t$-distribution with one degree of freedom.

Then $X$ has probability density function:

$\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$

where $k = 1$.

Hence:

\(\ds \map {f_X} x\) \(=\) \(\ds \dfrac {\map \Gamma {\frac {1 + 1} 2} } {\sqrt {\pi \times 1} \map \Gamma {\frac 1 2} } \paren {1 + \dfrac {x^2} 1}^{-\frac {1 + 1} 2}\) setting $k = 1$
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma 1} {\sqrt \pi \paren {1 + x^2} \map \Gamma {\frac 1 2} }\) simplifying
\(\ds \) \(=\) \(\ds \dfrac 1 {\sqrt \pi \paren {1 + x^2} \map \Gamma {\frac 1 2} }\) Gamma Function Extends Factorial: $\map \Gamma 1 = 1$
\(\ds \) \(=\) \(\ds \dfrac 1 {\pi \paren {1 + x^2} }\) Gamma Function of One Half: $\map \Gamma {\frac 1 2} = \sqrt \pi$

which is exactly the standard Cauchy distribution.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Student%27s_t-Distribution_with_One_Degree_of_Freedom_is_Standard_Cauchy_Distribution&oldid=654613"