32  Notable relations between distributions

32.1 Chi squared

The chi2 distribution with \(\nu\) degrees of freedom, namely \(\chi^2(\nu)\), is defined as the sum of \(\nu\)-independent and identically distributed standard normal random variables squared for \(i = 1, \dots, \nu\), i.e.  \[ \begin{cases} Z_i \sim \mathcal{N}(0,1) \quad \forall i \\ Z_i \perp Z_j \quad\quad\quad\; \forall i \neq j \end{cases} \implies X = Z_1^2 + \dots + Z_{\nu}^2 \sim \chi^{2}(\nu) \tag{32.1}\] The chi-squared distribution \(\chi^{2}(\nu)\) is a special case of the gamma distribution, i.e.  \[ X \sim \chi^{2}(\nu) \iff X \sim \text{Gamma}\left(\alpha = \frac{\nu}{2}, \theta = 2\right) \]

Expectation Variance Skewness Excess Kurtosis
\(\nu\) \(2 \nu\) \(\sqrt{\frac{8}{\nu}}\) \(\frac{12}{\nu}\)
Table 32.1: Moments of a \(\chi^2\) random variable
Sum of \(\chi^2\) random variables

The sum of two \(\chi^{2}\) is again \(\chi^{2}\) if and only if \(\chi^{2}(\nu_1)\) and \(\chi^{2}(\nu_2)\) are independent, formally:
\[ \chi^{2}(\nu_1) \perp \chi^{2}(\nu_2) \implies \chi^{2}(\nu_1) + \chi^{2}(\nu_2) \sim \chi^{2}(\nu_1 + \nu_2) \] If they are not independent their sum is not \(\chi^2(2)\) distributed.

32.1.1 Relations with others distributions

  1. \(\frac{1}{\nu} \chi^2 (\nu) \underset{\nu\to\infty}{\overset{\text{d}}{\longrightarrow}} 1\).
  2. \(\frac{\chi^2(\nu) - \nu}{\sqrt{2\nu}} \underset{\nu\to\infty}{\overset{\text{d}}{\longrightarrow}} \mathcal{N}(0,1)\).
  3. If \(\mathbf{x} \sim \mathcal{N}(\mathbf{0}, \boldsymbol{\Sigma})\) then \[ \mathbf{x}^{\small T} \boldsymbol{\Sigma}^{-1} \mathbf{x} \sim \chi^2(k) \text{.} \]
  4. A generalization of property 3. to non-central distributions: if \(\mathbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) then \[ \mathbf{x}^{\small T} \boldsymbol{\Sigma}^{-1} \mathbf{x} \sim \chi^2(k, \delta) \text{,} \quad \delta = \boldsymbol{\mu}^{\small T} \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu} \text{.} \]

32.2 Student-t

The Student-t distribution with \(\nu\) degrees of freedom, namely \(t(\nu)\), is defined as the ratio of two independent random variables. In specific, a standard normal random variable \(Z\) and the square root of a \(\chi^2(\nu)\) divided by its degrees of freedom \(\nu\), i.e.  \[ \begin{cases} Z \sim \mathcal{N}(0,1) \\ V \sim \chi^2(\nu) \\ Z \perp V \end{cases} \implies X = \frac{\sqrt{\nu} Z}{\sqrt{V}} \sim t(\nu) \tag{32.2}\] Given a location parameter \(\mu\) and a scale parameter \(\sigma^2\) the Student-t random variable admits the following stochastic representation: \[ X = \mu + \sigma \frac{\sqrt{\nu} \, Y}{\sqrt{V}} \sim t(\mu, \sigma^2, \nu) \]

Expectation Variance Skewness Excess Kurtosis
\(\mu\) \(\frac{\nu}{\nu-2} \sigma^2\) \(0\) \(\frac{6}{\nu - 4}, \; \nu > 4\)
Table 32.2: Moments of a Student-t random variable \(X \sim t(\mu, \sigma, \nu)\).

32.2.1 Relations with others distributions

  1. \(t(\nu) \underset{\nu\to\infty}{\overset{\text{d}}{\longrightarrow}} \mathcal{N}(0,1)\).

32.3 Fisher–Snedecor

The Fisher–Snedecor distribution with \(\nu_1\) and \(\nu_2\) degrees of freedom, often denoted as F, is defined as the ratio of two independent chi2 random variables, each one divided by its degrees of freedom, i.e.  \[ \begin{cases} V_1 \sim \chi^{2}(\nu_1) \\ V_2 \sim \chi^{2}(\nu_2) \end{cases} \implies X = \frac{\frac{V_1}{\nu_1}}{\frac{V_2}{\nu_2}} \sim \text{F}(\nu_1, \nu_2) \tag{32.3}\]

Expectation Variance Skewness Excess Kurtosis
\(\frac{\nu_2}{\nu_2 - 2}\) \(\frac{2 \nu_2^2 (\nu_1 + \nu_2 - 2)}{\nu_1 (\nu_2 - 2)^2 (\nu_2 -4)}\; \nu_2 > 2\) \(\frac{(2\nu_1 + \nu_2 - 2) \sqrt{8(\nu_2 - 4)}}{(\nu_2-6)\sqrt{\nu_1 (\nu_1 + \nu_2 - 2)}}\; \nu_2 > 6\) \(\frac{\nu_1 (5 \nu_2 - 22)(\nu_1 + \nu_2 - 2) + (\nu_2 - 4)(\nu_2 - 2)^2}{\nu_1( \nu_2 - 6) (\nu_2 - 8) (\nu_1 + \nu_2 - 2)}, \; \nu_2 > 8\)
Table 32.3: Moments of a \(\text{F}\) random variable

32.3.1 Relations with others distributions

  1. \(\nu_2 \, \text{F}(\nu_1, \nu_2) \underset{\nu_2\to\infty}{\overset{\text{d}}{\longrightarrow}} \chi^{2}(\nu_1)\).
  2. If \(X \sim \text{F}(\nu_1, \nu_2)\) then \(X^{-1} \sim \text{F}(\nu_2, \nu_1)\).
  3. If \(X \sim \text{t}(\nu)\), then \(X^{2} \sim \text{F}(1, \nu)\) and \(X^{-2} \sim \text{F}(\nu, 1)\).