Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ t-distribution ▷ Relationship to non-standardized t-distribution

Theorem: Let $X$ be a random variable following a non-standardized t-distribution with mean $\mu$, scale $\sigma^2$ and degrees of freedom $\nu$:

\[\label{eq:X} X \sim \mathrm{nst}(\mu, \sigma^2, \nu) \; .\]

Then, subtracting the mean and dividing by the square root of the scale results in a random variable following a t-distribution with degrees of freedom $\nu$:

\[\label{eq:nst-t} Y = \frac{X-\mu}{\sigma} \sim t(\nu) \; .\]

Proof: The non-standardized t-distribution is a special case of the multivariate t-distribution in which the mean vector and scale matrix are scalars:

\[\label{eq:nst-mvt} X \sim \mathrm{nst}(\mu, \sigma^2, \nu) \quad \Rightarrow \quad X \sim t(\mu, \sigma^2, \nu) \; .\]

Therefore, we can apply the linear transformation theorem for the multivariate t-distribution for an $n \times 1$ random vector $x$:

\[\label{eq:mvt-ltt} x \sim t(\mu, \Sigma, \nu) \quad \Rightarrow \quad y = Ax + b \sim t(A\mu + b, A \Sigma A^\mathrm{T}, \nu) \; .\]

Comparing with equation \eqref{eq:nst-t}, we have $A = 1/\sigma$, $b = -\mu/\sigma$ and the variable $Y$ is distributed as:

\[\label{eq:Y-dist} \begin{split} Y &= \frac{X-\mu}{\sigma} = \frac{X}{\sigma} - \frac{\mu}{\sigma} \\ &\sim t\left( \frac{\mu}{\sigma} - \frac{\mu}{\sigma}, \left( \frac{1}{\sigma} \right)^2 \sigma^2, \nu \right) \\ &= t\left( 0, 1, \nu \right) \; . \end{split}\]

Plugging $\mu = 0$, $\Sigma = 1$ and $n = 1$ into the probability density function of the multivariate t-distribution,

\[\label{eq:mvt-pdf} p(x) = \sqrt{\frac{1}{(\nu \pi)^{n} |\Sigma|}} \, \frac{\Gamma([\nu+n]/2)}{\Gamma(\nu/2)} \, \left[ 1 + \frac{1}{\nu} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right] \; ,\]

we get

\[\label{eq:t-pdf} p(x) = \sqrt{\frac{1}{\nu \pi}} \, \frac{\Gamma([\nu+1]/2)}{\Gamma(\nu/2)} \, \left[ 1 + \frac{x^2}{\nu} \right]\]

which is the probability density function of Student’s t-distribution with $\nu$ degrees of freedom.

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Metadata: ID: P232 | shortcut: nst-t | author: JoramSoch | date: 2021-05-11, 15:46.