Recall, that for the univariate (one random variable) situation: Given \(X\) with pdf \(f(x)\) and the transformation \(Y=u(X)\) with the single-valued inverse \(X=v(Y)\), then the pdf of \(Y\) is given by

\(\begin{align*} g(y) = |v^\prime(y)| f\left[ v(y) \right]. \end{align*}\)

Now, suppose \((X_1, X_2)\) has joint density \(f(x_1, x_2)\). and support \(S_X\).

Let \((Y_1, Y_2)\) be some function of \((X_1, X_2)\) defined by \(Y_1 = u_1(X_1, X_2)\) and \(Y_2 = u_2(X_1, X_2)\) with the single-valued inverse given by \(X_1 = v_1(Y_1, Y_2)\) and \(X_2 = v_2(Y_1, Y_2)\). Let \(S_Y\) be the support of \(Y_1, Y_2\).

Then, we usually find \(S_Y\) by considering the image of \(S_X\) under the transformation \((Y_1, Y_2)\). Say, given \(x_1, x_2 \in S_X\), we can find \((y_1, y_2) \in S_Y\) by

\(\begin{align*} x_1 = v_1(y_1, y_2), \hspace{1cm} x_2 = v_2(y_1, y_2) \end{align*}\)

The joint pdf \(Y_1\) and \(Y_2\) is

\(\begin{align*} g(y_1, y_2) = |J| f\left[ v_1(y_1, y_2), v_2(y_1, y_2) \right] \end{align*}\)

In the above expression, \(|J|\) refers to the absolute value of the Jacobian, \(J\). The Jacobian, \(J\), is given by

\(\begin{align*} \left| \begin{array}{cc} \frac{\partial v_1(y_1, y_2)}{\partial y_1} & \frac{\partial v_1(y_1, y_2)}{\partial y_2} \\ \frac{\partial v_2(y_1, y_2)}{\partial y_1} & \frac{\partial v_2(y_1, y_2)}{\partial y_2} \end{array} \right| \end{align*}\)

i.e. it is the determinant of the matrix

\(\begin{align*} \left( \begin{array}{cc} \frac{\partial v_1(y_1, y_2)}{\partial y_1} & \frac{\partial v_1(y_1, y_2)}{\partial y_2} \\ \frac{\partial v_2(y_1, y_2)}{\partial y_1} & \frac{\partial v_2(y_1, y_2)}{\partial y_2} \end{array} \right) \end{align*}\)

Let \(X_1\) and \(X_2\) have independent gamma distributions with parameters \(\alpha, \theta\) and \(\beta\) respectively. Therefore, the joint pdf of \(X_1\) and \(X_2\) is given by

\(\begin{align*} f(x_1, x_2) = \frac{1}{\Gamma(\alpha) \Gamma(\beta)\theta^{\alpha + \beta}} x_1^{\alpha-1}x_2^{\beta-1}\text{ exp }\left( -\frac{x_1 + x_2}{\theta} \right), 0 <x_1 <\infty, 0 <x_2 <\infty. \end{align*}\)

We make the following transformation:

\(\begin{align*} Y_1 = \frac{X_1}{X_1+X_2}, Y_2 = X_1+X_2 \end{align*}\)

The inverse transformation is given by

\(\begin{align*} &X_1=Y_1Y_2, \\& X_2=Y_2-Y_1Y_2 \end{align*}\)

The Jacobian is

\(\begin{align*} \left| \begin{array}{cc} y_2 & y_1 \\ -y_2 & 1-y_1 \end{array} \right| = y_2(1-y_1) + y_1y_2 = y_2 \end{align*}\)

The joint pdf \(g(y_1, y_2)\) is

\(\begin{align*} g(y_1, y_2) = |y_2| \frac{1}{\Gamma(\alpha) \Gamma(\beta)\theta^{\alpha + \beta}} (y_1y_2)^{\alpha - 1}(y_2 - y_1y_2)^{\beta - 1}e^{-y_2/\theta} \end{align*}\)

with support is \(0<y_1<1, 0<y_2<\infty\)

It may be shown that the marginal pdf of \(Y_1\) is

\(\begin{align*} g(y_1) & = \frac{y_1^{\alpha - 1}(1 - y_1)^{\beta - 1}}{\Gamma(\alpha) \Gamma(\beta) } \int_0^{\infty} \frac{y_2^{\alpha + \beta -1}}{\theta^{\alpha + \beta}} e^{-y_2/\theta} dy_2 g(y_1) \\& = \frac{ \Gamma(\alpha + \beta) }{\Gamma(\alpha) \Gamma(\beta) } y_1^{\alpha - 1}(1 - y_1)^{\beta - 1}, \hspace{1cm} 0<y_1<1. \end{align*}\)

\(Y_1\) is said to have a beta pdf with parameters \(\alpha\) and \(\beta\).

We describe a very useful distribution in Statistics known as the F distribution.

Let \(U\) and \(V\) be independent chi-square variables with \(r_1\) and \(r_2\) degrees of freedom, respectively. The joint pdf is

\(\begin{align*}
g(u, v) = \frac{ u^{r_1/2-1}e^{-u/2} v^{r_2/2-1}e^{-v/2} } { \Gamma (r_1/2) 2^{r_1/2} \Gamma
(r_2/2) 2^{r_2/2} } , \hspace{1cm} 0<u<\infty, 0<v<\infty
\end{align*}\)

Define the random variable \(W = \frac{U/r_1}{V/r_2}\)

This time we use the distribution function technique described in lesson 22,

\(\begin{align*}
F(w) = P(W \leq w)
= P \left( \frac{U/r_1}{V/r_2} \leq w \right) = P(U \leq \frac{r_1}{r_2} wV) = \int_0^\infty \int_0^{(r_1/r_2)wv} g
(u, v) du dv
\end{align*}\)

\(\begin{align*}
F(w) =\frac{1}{ \Gamma (r_1/2) \Gamma (r_2/2) } \int_0^\infty \left[ \int_0^
{(r_1/r_2)wv} \frac{ u^{r_1/2-1}e^{-u/2}}{2^{(r_1+r_2)/2}} du \right] v^{r_1/2-1}e^{-v/2} dv
\end{align*}\)

By differentiating the cdf , it can be shown that \(f(w) = F^\prime(w)\) is given by

\(\begin{align*}
f(w) = \frac{ \left( r_1/r_2 \right)^{r_1/2} \Gamma \left[ \left(r_1+r_2\right)/2 \right]w^{r_1/2-1} }
{\Gamma(r_1/2)\Gamma(r_2/2) \left[1+(r_1w/r_2)\right]^{(r_1+r_2)/2}}, \hspace{1cm} w>0
\end{align*}\)

A random variable with the pdf \(f(w)\) is said to have an F distribution with \(r_1\) and \(r_2\) degrees of freedom. We write this as \(F(r_1, r_2)\). Table VII in Appendix B of the textbook can be used to find probabilities for a random variable with the \(F(r_1, r_2)\) distribution.

It contains the F-values for various cumulative probabilities \((0.95, 0.975, 0.99)\) (or the equivalent upper − \(\alpha\)th probabilities \((0.05, 0.025, 0.01)\)) of various \(F (r1, r2)\) distributions.

When using this table, it is helpful to note that if a random variable (say, \(W\)) has the \(F(r_1, r_2)\) distribution, then its inverse \(\dfrac{1}{W}\) has the \(F(r_2, r_1)\) distribution.

The shape of the F distribution is determined by the degrees of freedom \(r_1\) and \(r_2\). The histogram below shows how an F random variable is generated using 1000 observations each from two chi-square random variables (\(U\) and \(V\)) with degrees of freedom 4 and 8 respectively and forming the ratio \(\dfrac{U/4}{V/8}\).

The lower plot (below histogram) illustrates how the shape of an F distribution changes with the degrees of freedom \(r_1\) and \(r_2\).