General Linear Transformation

Here we will use matrix notation to express the results of 3.3, and give a useful result using moment generating functions.

The one-to-one linear transformation referred to in section 3.3 on page  can be written in matrix notation as ${\bf Y} = {\bf AX}$, (using ${\bf Y}$ for the new variables rather that ${\bf U}$). Here ${\bf X}$ and ${\bf Y}$ are vectors of random variables and ${\bf A}$ is a matrix of constants. In particular, note that $E({\bf X})$ is the vector whose components are E(X$_1$), ... E(X$_p$), or $\mu_1, \ldots, \mu_p$. The covariance matrix of ${\bf X}$ (sometimes called the variance-covariance matrix) is frequently referred to as cov$({\bf X})$, and is denoted by $\bf\Sigma$. Note that it is a square matrix whose diagonal terms are variances, and off-diagonal terms are covariances.

If ${\bf A}$ is non-singular so that there is an inverse transformation ${\bf X}={\bf A}^{-1}{\bf Y}$, and if ${\bf X}$ has pdf $f_{\bf X}({\bf x})$, the corresponding pdf of ${\bf Y}$ is

$$f_{\bf Y}({\bf y}) \, = \, f_{\bf X}({\bf A}^{-1}{\bf y})\mb... ..._{\bf X}({\bf A}^{-1}{\bf y})\mbox{abs}\vert{\bf A}^{-1}\vert.$$ (3.6)

Recall that the joint mgf of $(X_1,\,X_2,\,\dots ,\, X_p)$ is expressed in matrix notation as

$$M_{\bf X}({\bf t}) = E\left(e^{t_1X_1+t_2X_2+\dots +t_pX_p}\right) = E(e^{{\bf t}'{\bf X}})$$

provided this expectation exists. Now if ${\bf Y} = {\bf AX}$, so that ${\bf Y}$ is a p-dimensional random vector, the mgf of ${\bf Y}$ is

$$M_{\bf Y}({\bf t})=E(e^{{\bf t}'{\bf Y}})=E(e^{{\bf t}'{\bf... ...e^{({\bf A}'{\bf t})'{\bf X}}) = M_{\bf X} ({\bf A}'{\bf t})$$ (3.7)