How to Derive the F Distribution

Math Proof, Statistics / By /
f-distribution from chi-squared distribution

Table of Contents


Context


F Distribution Definition

The F Distribution is the ratio of two independent Chi-squared distributions standardized by their degrees of freedom.
\[\large{
\begin{align}
Y_k\sim\chi^2(k) && X\sim F(p,q) = \frac{Y_p/p}{Y_q/q}
\end{align}
}\]


Single Variable Change

Similar to the T Distribution, deriving the F Distribution involves several single variable changes.

A variable change is a change to a functions variable and domain such that the new and old functions have integrals which are equal when evaluated over their respective domains (eg if you perform a variable change on a probability function you would still expect the area under the probability curve to be one).
\[\large{
\begin{align}
y = g(x) && \int_y f(y) \partial y = \int_x f(g(x)) \frac{\partial g}{\partial x}\partial x
\end{align}
}\]


Multi-variable Change with Jacobian Matrix

This proof involves one multi-variable change. In order to accomplish this the determinant of a Jacobian matrix is required.

**NOTE**: The combination of the Jacobian matrix and its determinant are often referred to together as simple the Jacobian.

A function with more than one scalar input variable can also be expressed as a function with a single vector input variable.
\[\large{
f(y_1, y_2, y_3) = f\Big(\stackrel{y}{\begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix}}\Big) = f(y)
}\]
A multi-variable change requires that the input vector for the original function can itself be expressed as the range of a vector valued function of the variable you wish to change to.
\[\large{
\begin{align}
\stackrel{y}{\begin{bmatrix}y_1\\ y_2 \end{bmatrix}} = g\Big(\stackrel{x}{\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}}\Big) = \stackrel{g(x)}{\begin{bmatrix} x_1 + x_2 + x_3 \\ 5x_1 x_3 \end{bmatrix}}
\end{align}
}\]
A Jacobian matrix is a matrix of the first order partial derivatives of a vector valued function with respect to its vector input domain.
\[\large{
\stackrel{g(x)}{\begin{bmatrix} g_1(x_1,\ldots,x_n) \\ \vdots \\ g_k(x_1,\ldots,x_n) \end{bmatrix}}\longrightarrow J_{g(x)} = \begin{bmatrix}
\frac{\partial g_1}{\partial x_1} & \ldots & \frac{\partial g_1}{\partial x_n} \\
\vdots & \ddots & \vdots \\
\frac{\partial g_k}{\partial x_1} & \ldots & \frac{\partial g_k}{\partial x_n}
\end{bmatrix}
}\]
If the determinant of the Jacobian matrix exists it can be used to perform a multi-variable change on functions where the input vector is the domain of the function the Jacobian matrix was built from.
\[\large{
\begin{align}
y = g(x) && \int_y f(y) \partial y = \int_x f(g(x))\cdot \text{det}(J_{g(x)}) \partial x && \frac{\partial f}{\partial y} = \frac{\partial f}{\partial g} \text{det}(J_{g(x)})
\end{align}
}\]


How to Derive the F Distribution

An F random variable is defined as the ratio of two Chi-Squared random variables standardized by their own degrees of freedom.

Let \(X\) be the ratio of two independent Chi-squared random variables, \(Y_p\) and \(Y_q\), standardized by their degrees of freedom parameters \(p\) and \(q\) respectively. Then \(X\) is an F random variable with density function for \(f_X(x;p,q) = \frac{(x)^{p/2-1}p^{p/2}q^{q/2}}{\text{B}(p/2,q/2)(xp+q)^{p/2+q/2}}\).

First declare two related equalities (because they share a variable).
\[\large{
\begin{align}
&W_1 = Y_p/Y_q& &\longrightarrow& &Y_p = W_1 W_2 && \normalsize (1) \\ \\
&W_2 = Y_q& &\longrightarrow& &Y_q = W_2 \\ \\
\end{align}
}\]
The above equalities can be expressed in vector form as below and vector y can be expressed as a function of vector w.
\[\large{
\begin{align}
\stackrel{w}{\begin{bmatrix} W_1 \\ W_2\end{bmatrix}} && \stackrel{y}{\begin{bmatrix} Y_p \\ Y_q \end{bmatrix}} = \stackrel{g(w)}{\begin{bmatrix} W_1W_2 \\ W_2 \end{bmatrix}} && \normalsize (2) \\ \\
\end{align}
}\]
\(Y_p\) and \(Y_q\) are independent random variables so their joint probability function is simply the product of their individual Chi-squared probability functions.
\[\large{
\begin{align}
f_{Y_p,Y_q}(y_p,y_q) &= \frac{y_p^{p/2-1}e^{-y_p/2}}{\Gamma(p/2)2^{p/2}}\cdot\frac{y_q^{q/2-1}e^{-y_q/2}}{\Gamma(q/2)2^{q/2}} && \normalsize (3) \\ \\
&= \frac{y_p^{p/2-1}y_q^{q/2-1}e^{-\frac{(y_p + y_q)}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}
\end{align}
}\]
The next step is to perform a multi-variable change on the joint probability function from \(y\) to \(w\) using the Jacobian (instructions above).
\[\large{
\begin{align}
&J_{g(w)} = \begin{bmatrix} \frac{\partial g_1}{\partial w_1} & \frac{\partial g_1}{\partial w_2} \\
\frac{\partial g_2}{\partial w_1} & \frac{\partial g_2}{\partial w_2}
\end{bmatrix} =\begin{bmatrix} w_2 & w_1 \\
0 & 1
\end{bmatrix} && \normalsize (4)\\ \\
&\text{det}(J_{g(w)}) = w_2
\end{align}
}\]
Express the integral of the joint probability function \(Y_p\) and \(Y_q\) as a function of vector \(y\), then perform the variable change from \(y\) to \(w\) using the Jacobian.
\[\require{\cancel}\large{
\begin{align}
f(y) &= f(g(w))\cdot\text{det}(J_{g(w)}) &&\normalsize (5) \\ \\

&= \frac{(w_1w_2)^{p/2-1}w_2^{q/2\cancel{-1}}e^{-\frac{(w_1w_2 + w_2)}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot \cancel{w_2} \\ \\

f_{W_1,W_2}(w_1,w_2;p,q)&= \frac{w_1^{p/2-1}w_2^{p/2+q/2-1}e^{-\frac{(w_1w_2 + w_2)}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\partial w
\end{align}
}\]
Now integrate out random variable \(W_2\) to obtain the univariate probability function of \(W_1\).

In order to do this notice that now there is a potential gamma function kernel given by \(w_2^{p/2+q/2-1}e^{-w_2\frac{w_1+1}{2}}\). In order to complete the expression of the gamma function two steps are necessary. First insert multiply the function by 1 and factor it: \(\require{\color}{\color{aqua}1 = \Big(\frac{2(w_1+1)}{2(w_1+1)}\Big)^{p/2+q/2-1}}\).
\[\large{
\begin{align}
\int_0^\infty f_{W_1,W_2}(w_2;p,q)\partial w_2 &= \int_0^\infty \frac{w_1^{p/2-1}w_2^{p/2+q/2-1}e^{-w_2\frac{w_1 + 1}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot{\color{aqua}\Big(\frac{2(w_1+1)}{2(w_1+1)}\Big)^{p/2+q/2-1}} \partial w_2 &&\normalsize (6) \\ \\

&= \int_0^\infty \frac{w_1^{p/2-1}(w_2{\color{aqua}\frac{w_1+1}{2}})^{p/2+q/2-1}e^{-w_2\frac{w_1 + 1}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot{\color{aqua}\Big(\frac{2}{w_1+1}\Big)^{p/2+q/2-1}} \partial w_2
\end{align}
}\]
Step two to completing the gamma function expression is to perform another variable change, this time from \(W_2\) to \(t\). Let \(t = w_2\frac{w_1+1}{2}\), which implies \(W_2 = g(t) = \frac{2t}{w_1+1}\).
\[\large{
\begin{align}
\int_0^\infty f_{W_1,W_2}(w_2) \partial w_2 &= \int_0^\infty f_{W_1,W_2}(g(t))\cdot\frac{\partial g}{\partial t} \partial t &&\normalsize (7) \\ \\

&= {\color{gold}\int_0^\infty} \frac{w_1^{p/2-1}{\color{gold}t^{p/2+q/2-1}e^{-t}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot\Big(\frac{2}{w_1+1}\Big)^{p/2+q/2\cancel{-1}}\cdot\cancel{\frac{2}{w_1+1}}{\color{gold}\partial t} \\ \\

&= \frac{w_1^{p/2-1}{\color{gold}\Gamma(p/2+q/2)}}{\Gamma(p/2)\Gamma(q/2)\cancel{2^{p/2+q/2}}}\cdot\Big(\frac{\cancel{2}}{w_1+1}\Big)^{p/2+q/2}
\end{align}
}\]
After obtaining \(\Gamma(p/2+q/2)\) it can be combined with the other two gamma functions into a beta function.
\[\large{
\begin{align}
f_{W_1}(w_1;p,q) &= \frac{w_1^{p/2-1}}{\text{B}(p/2,q/2)}\cdot\Big(\frac{1}{w_1+1}\Big)^{p/2+q/2} && \normalsize (8)
\end{align}
}\]
Recall from (1) that \(W_1\) is the unstandardized ratio random variable of two Chi-squared random variables. Let \(X = W_1\frac{q}{p}\), which makes it the standardized ratio of two Chi-squared random variables. Perform a final variable change from \(W_1\) to \(X\).
\[\large{
\begin{align}
f_{W_1}(w_1;p,q) &= f_{W_1}(g(x);p,q)\cdot\frac{\partial g}{\partial x} &&\normalsize(9) \\ \\
f_X(x;p,q) &= \frac{x^{p/2-1}(\frac{p}{q})^{p/2\cancel{-1}}}{\text{B}(p/2,q/2)}\cdot\Big(\frac{1}{x\frac{p}{q}+1}\Big)^{p/2+q/2}\cdot \cancel{\frac{p}{q}} \\ \\
&= \frac{x^{p/2-1}(\frac{p}{\cancel{q}})^{p/2}}{\text{B}(p/2,q/2)}\cdot\frac{q^{\cancel{p/2}+q/2}}{(xp+q)^{p/2+q/2}}\\ \\
&= \frac{x^{p/2-1}p^{p/2}q^{q/2}}{\text{B}(p/2,q/2)(xp+q)^{p/2+q/2}} &&\normalsize (\text{QED})
\end{align}
}\]


Summary Conclusion

Deriving the F distribution probability function from the Chi-squared distribution isn’t a complicated task, but it can get messy with 3 variable changes.

Start from the definition of an F random variable defined as the ratio of two independent Chi-squared random variables scaled by their degrees of freedom. Let the two Chi-squared be \(Y_p\) and \(Y_q\). Define two more random variables via two equivalencies \(W_1 = \frac{Y_p}{Y_q}Y_p = Y_p\quad W_2=Y_q\).

Use the equivalencies between the y and w random to perform a multi-variable change on the joint probability function \(f_Y(y)\). Next integrate out \(W_2\) by completing a gamma function kernel. This leaves the single variable probability function for \(W_1\).

\(W_1\) is defined as the unstandardized ratio random variable of two independent Chi-squared random variables. Perform a final variable change, simply applying the standardizing scalar to the random variable, to arrive at the probability function for an F random variable.

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