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Derivation of Beta Distribution from F-Distribution 📂Probability Distribution

Derivation of Beta Distribution from F-Distribution

Theorem 1

A random variable XF(r1,r2)X \sim F \left( r_{1}, r_{2} \right) following an F-distribution with degrees of freedom r1,r2r_{1} , r_{2} is defined as follows YY and follows a beta distribution Best(r12,r22)\text{Best} \left( {{ r_{1} } \over { 2 }} , {{ r_{2} } \over { 2 }} \right). Y:=(r1/r2)X1+(r1/r2)XBeta(r12,r22) Y := {{ \left( r_{1} / r_{2} \right) X } \over { 1 + \left( r_{1} / r_{2} \right) X }} \sim \text{Beta} \left( {{ r_{1} } \over { 2 }} , {{ r_{2} } \over { 2 }} \right)

Proof

Strategy: Direct deduction using the probability density function.

Definition of F-distribution: A continuous probability distribution F(r1,r2)F \left( r_{1} , r_{2} \right) with the following probability density function for degrees of freedom r1,r2>0r_{1}, r_{2} > 0 is called an F-distribution. f(x)=1B(r1/2,r2/2)(r1r2)r1/2xr1/21(1+r1r2x)(r1+r2)/2,x(0,) f(x) = {{ 1 } \over { B \left( r_{1}/2 , r_{2} / 2 \right) }} \left( {{ r_{1} } \over { r_{2} }} \right)^{r_{1} / 2} x^{r_{1} / 2 - 1} \left( 1 + {{ r_{1} } \over { r_{2} }} x \right)^{-(r_{1} + r_{2}) / 2} \qquad , x \in (0, \infty)

Definition of Beta Distribution: A continuous probability distribution Beta(α,β)\text{Beta}(\alpha,\beta) with the following probability density function for α,β>0\alpha , \beta > 0 is called a Beta distribution. f(x)=1B(α,β)xα1(1x)β1,x[0,1] f(x) = {{ 1 } \over { B(\alpha,\beta) }} x^{\alpha - 1} (1-x)^{\beta - 1} \qquad , x \in [0,1]

  • B(r1/2,r2/2)B(r_{1} / 2, r_{2}/2) signifies the Beta function.

Y=(r1/r2)X1+(r1/r2)X    Y(1+(r1/r2)X)=(r1/r2)X    Y=(r1/r2)X(1Y)    (r1/r2)X=Y1Y \begin{align*} & Y = {{ \left( r_{1} / r_{2} \right) X } \over { 1 + \left( r_{1} / r_{2} \right) X }} \\ \implies & Y \left( 1 + \left( r_{1} / r_{2} \right) X \right) = \left( r_{1} / r_{2} \right) X \\ \implies & Y = \left( r_{1} / r_{2} \right) X (1 - Y) \\ \implies & \left( r_{1} / r_{2} \right) X = {{ Y } \over { 1 - Y }} \end{align*} and dy=[(r1/r2)1+(r1/r2)x(r1/r2)(r1/r2)x[1+(r1/r2)x]2]dx=(r1/r2)1+(r1/r2)x[1+(r1/r2)x1+(r1/r2)x(r1/r2)x1+(r1/r2)x]dx=(r1/r2)[1+(r1/r2)x]2dx \begin{align*} dy =& \left[ {{ \left( r_{1} / r_{2} \right) } \over { 1 + \left( r_{1} / r_{2} \right) x }} - \left( r_{1} / r_{2} \right) {{ \left( r_{1} / r_{2} \right) x } \over { \left[ 1 + \left( r_{1} / r_{2} \right) x \right]^{2} }} \right] dx \\ =& {{ \left( r_{1} / r_{2} \right) } \over { 1 + \left( r_{1} / r_{2} \right) x }} \left[ {{ 1 + \left( r_{1} / r_{2} \right) x } \over { 1 + \left( r_{1} / r_{2} \right) x }} - {{ \left( r_{1} / r_{2} \right) x } \over { 1 + \left( r_{1} / r_{2} \right) x }} \right] dx \\ =& {{ \left( r_{1} / r_{2} \right) } \over { \left[ 1 + \left( r_{1} / r_{2} \right) x \right]^{2} }} dx \end{align*} therefore, the probability density function fYf_{Y} of YY is B(r1/2,r2/2)fY(y)=(r1r2)r1/2xr1/21(1+r1r2x)(r1+r2)/2[1+(r1/r2)x]2(r1/r2)=(r1r2)r1/21xr1/21(1+r1r2x)2(r1+r2)/2=(r1r2x)r1/21(1+r1r2x)2(r1+r2)/2=yr1/21(1+r1r2x)r1/21(1+r1r2x)2(r1+r2)/2=yr1/21(1+r1r2x)1r2/2=yr1/21(1+y1y)1r2/2=yr1/21(11y)1r2/2=yr1/21(1y)r2/21 \begin{align*} & B \left( r_{1}/2 , r_{2} / 2 \right) f_{Y} (y) \\ =& \left( {{ r_{1} } \over { r_{2} }} \right)^{r_{1} / 2} x^{r_{1} / 2 - 1} \left( 1 + {{ r_{1} } \over { r_{2} }} x \right)^{-(r_{1} + r_{2}) / 2} \cdot {{ \left[ 1 + \left( r_{1} / r_{2} \right) x \right]^{2} } \over { \left( r_{1} / r_{2} \right) }} \\ =& \left( {{ r_{1} } \over { r_{2} }} \right)^{r_{1} / 2 - 1} x^{r_{1} / 2 - 1} \left( 1 + {{ r_{1} } \over { r_{2} }} x \right)^{2-(r_{1} + r_{2}) / 2} \\ =& \left( {{ r_{1} } \over { r_{2} }} x \right)^{r_{1} / 2 - 1} \left( 1 + {{ r_{1} } \over { r_{2} }} x \right)^{2-(r_{1} + r_{2}) / 2} \\ =& y^{r_{1} / 2 - 1} \left( 1 + {{ r_{1} } \over { r_{2} }} x \right)^{r_{1} / 2 - 1} \left( 1 + {{ r_{1} } \over { r_{2} }} x \right)^{2-(r_{1} + r_{2}) / 2} \\ =& y^{r_{1} / 2 - 1} \left( 1 + {{ r_{1} } \over { r_{2} }} x \right)^{1 - r_{2} / 2} \\ =& y^{r_{1} / 2 - 1} \left( 1 + {{ y } \over { 1 - y }} \right)^{1 - r_{2} / 2} \\ =& y^{r_{1} / 2 - 1} \left( {{ 1 } \over { 1 - y }} \right)^{1 - r_{2} / 2} \\ =& y^{r_{1} / 2 - 1} \left( 1 - y \right)^{r_{2} / 2 - 1} \end{align*} Summarizing, YY has the following probability density function Beta(r12,r22)\text{Beta} \left( {{ r_{1} } \over { 2 }} , {{ r_{2} } \over { 2 }} \right). fY(y)=1B(r1/2,r2/2)yr1/21(1y)r2/21 f_{Y} (y) = {{ 1 } \over { B \left( r_{1}/2 , r_{2} / 2 \right) }} y^{r_{1} / 2 - 1} \left( 1 - y \right)^{r_{2} / 2 - 1}


  1. Casella. (2001). Statistical Inference(2nd Edition): p225. ↩︎