Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Beta distribution ▷ Relationship to F-distribution

Theorem: Let $X$ be a random variable following an F-distribution:

\[\label{eq:X} X \sim F(d_1, d_2) \; .\]

Then, the quantity

\[\label{eq:Y} Y = \frac{d_1 X / d_2}{1 + d_1 X / d_2} = \frac{d_1 X}{d_2 + d_1 X}\]

follows a beta distribution:

\[\label{eq:beta-f} Y \sim \mathrm{Bet}\left( \frac{d_1}{2}, \frac{d_2}{2} \right) \; .\]

Proof: We denote $Y = g(X)$. The first derivative of the $g$ is:

\[\label{eq:dg-dx} \begin{split} \frac{\mathrm{d}g(x)}{\mathrm{d}x} &= \frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{d_1 x}{d_2 + d_1 x} \right) \\ &= \frac{d_1 d_2 + d_1^2 x - d_1^2 x}{(d_2 + d_1 x)^2} \\ &= \frac{d_1 d_2}{(d_2 + d_1 x)^2} \; . \end{split}\]

This derivative is positive for all $x \geq 0$, such that $Y$ is a strictly increasing function of $X$. This means we can derive the distribution of $Y$ by applying the probability density function of a strictly increasing function of a continuous random variable:

\[\label{eq:pdf-sifct} f_Y(y) = \left\{ \begin{array}{rl} f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; , & \text{if} \; y \in \mathcal{Y} \\ 0 \; , & \text{if} \; y \notin \mathcal{Y} \; . \end{array} \right.\]

where $\mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace$. Because $g: \, \mathbb{R}_{\geq 0} \rightarrow [0,1]$, we have $\mathcal{X} = \mathbb{R}_{\geq 0}$ and $\mathcal{Y} = [0, 1]$, such that

\[\label{eq:Y-pdf-zero} f_Y(y) = 0 \quad \text{for} \quad y \notin [0,1] \; .\]

The inverse function of $g$ is:

\[\label{eq:g-inv} \begin{split} g(x) = y &= \frac{d_1 x}{d_2 + d_1 x} \\ d_2 y + d_1 x y &= d_1 x \\ d_1 x - d_1 x y &= d_2 y \\ x (d_1 - d_1 y) &= d_2 y \\ x &= \frac{d_2 y}{d_1 - d_1 y} = g^{-1}(y) \; . \end{split}\]

The derivative of $g^{-1}$ with respect to $y$ is:

\[\label{eq:dg-inv-dy} \begin{split} \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} &= \frac{\mathrm{d}}{\mathrm{d}y} \left( \frac{d_2 y}{d_1 - d_1 y} \right) \\ &= \frac{d_1 d_2 + d_1 d_2 y - (- d_1 d_2 y)}{(d_1 - d_1 y)^2} \\ &= \frac{d_1 d_2}{d_1^2 - 2 d_1^2 y + d_1^2 y^2} \\ &= \frac{d_2}{d_1 - 2 d_1 y + d_1 y^2} \\ &= \frac{d_2}{d_1 (1 - 2 y + y^2)} \\ &= \frac{d_2}{d_1} \cdot \frac{1}{(1-y)^2} \; . \end{split}\]

The probability density function of the F-distribution with degrees of freedom $d_1$ and $d_2$ is:

\[\label{eq:f-pdf} f_X(x) = \frac{\Gamma\left( \frac{d_1+d_2}{2} \right)}{\Gamma\left( \frac{d_1}{2} \right) \cdot \Gamma\left( \frac{d_2}{2} \right)} \cdot \left( \frac{d_1}{d_2} \right)^{\frac{d_1}{2}} \cdot x^{\frac{d_1}{2}-1} \cdot \left( \frac{d_1}{d_2} x + 1 \right)^{-\frac{d_2+d_2}{2}} \; .\]

With that, we have everything that we need to derive the distribution of $Y$ for $y \in \mathcal{Y}$. Combining \eqref{eq:pdf-sifct}, \eqref{eq:f-pdf}, \eqref{eq:g-inv} and \eqref{eq:dg-inv-dy}, $f_Y(y)$ for $y \in [0,1]$ becomes:

\[\label{eq:Y-pdf-nonzero} \begin{split} f_Y(y) &= f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \\ &= \frac{\Gamma\left( \frac{d_1+d_2}{2} \right)}{\Gamma\left( \frac{d_1}{2} \right) \cdot \Gamma\left( \frac{d_2}{2} \right)} \cdot \left( \frac{d_1}{d_2} \right)^{\frac{d_1}{2}} \cdot \left( \frac{d_2 y}{d_1 - d_1 y} \right)^{\frac{d_1}{2}-1} \cdot \left( \frac{d_1}{d_2} \left( \frac{d_2 y}{d_1 - d_1 y} \right) + 1 \right)^{-\frac{d_1+d_2}{2}} \cdot \\ &\hphantom{=} \left( \frac{d_2}{d_1} \cdot \frac{1}{(1-y)^2} \right) \\ &= \frac{1}{\mathrm{B}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)} \cdot \left( \frac{d_1}{d_2} \right)^{\frac{d_1}{2}} \cdot d_2^{\frac{d_1}{2}-1} \cdot y^{\frac{d_1}{2}-1} \cdot d_1^{-\left(\frac{d_1}{2}-1\right)} \cdot (1-y)^{-\left(\frac{d_1}{2}-1\right)} \cdot \\ &\hphantom{=} \left( \frac{d_1}{d_2} \left( \frac{d_2 y}{d_1 - d_1 y} \right) + 1 \right)^{-\frac{d_1+d_2}{2}} \cdot \left( \frac{d_2}{d_1} \cdot \frac{1}{(1-y)^2} \right) \\ &= \frac{1}{\mathrm{B}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)} \cdot \left( \frac{d_1}{d_2} \right) \cdot \left( \frac{d_2}{d_1} \right) \cdot y^{\frac{d_1}{2}-1} \cdot (1-y)^{-\frac{d_1}{2}+1} \cdot (1-y)^{-2} \cdot \\ &\hphantom{=} \left( \frac{d_1 d_2 y}{d_2 (d_1 - d_1 y)} + \frac{d_2 (d_1 - d_1 y)}{d_2 (d_1 - d_1 y)} \right)^{-\frac{d_1+d_2}{2}} \\ &= \frac{1}{\mathrm{B}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)} \cdot y^{\frac{d_1}{2}-1} \cdot (1-y)^{-\frac{d_1}{2}-1} \cdot \left( \frac{d_1 d_2}{d_2 (d_1 - d_1 y)} \right)^{-\frac{d_1+d_2}{2}} \\ &= \frac{1}{\mathrm{B}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)} \cdot y^{\frac{d_1}{2}-1} \cdot (1-y)^{-\frac{d_1}{2}-1} \cdot \left( \frac{1}{1 - y} \right)^{-\frac{d_1+d_2}{2}} \\ &= \frac{1}{\mathrm{B}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)} \cdot y^{\frac{d_1}{2}-1} \cdot (1-y)^{-\frac{d_1}{2}-1} \cdot (1-y)^{\frac{d_1+d_2}{2}} \\ &= \frac{1}{\mathrm{B}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)} \cdot y^{\frac{d_1}{2}-1} \cdot (1-y)^{\frac{d_2}{2}-1} \; . \end{split}\]

This is the probability density function of the beta distribution with parameters

\[\label{eq:beta-f-para} \alpha = \frac{d_1}{2} \quad \mathrm{and} \quad \beta = \frac{d_2}{2} \; ,\]

such that

\[\label{eq:beta-f-qed} Y \sim \mathrm{Bet}\left( \frac{d_1}{2}, \frac{d_2}{2} \right) \; .\]
∎
Sources:

Metadata: ID: P497 | shortcut: beta-f | author: JoramSoch | date: 2025-04-04, 14:42.