  With $<\cdot>$ denoting an average with respect to the eigenvalue PDF for the
Laguerre unitary ensemble, the object of our study is $ \tilde{E}_N(I;a,\mu) :=
< \prod_{l=1}^N \chi_{(0,\infty)\backslash I}^{(l)} (\lambda - \lambda_l)^\mu>$
for $I = (0,s)$ and $I = (s,\infty)$, where $\chi_I^{(l)} = 1$ for $ \lambda_l
\in I$ and $\chi_I^{(l)} = 0$ otherwise. Using Okamoto's development of the
theory of the Painlev\'e V equation, it is shown that $\tilde{E}_N(I;a,\mu)$ is
a $\tau$-function associated with the Hamiltonian therein, and so can be
characterised as the solution of a certain second order second degree
differential equation, or in terms of the solution of certain difference
equations. The cases $\mu = 0$ and $\mu = 2$ are of particular interest as they
correspond to the cumulative distribution and density function respectively for
the smallest and largest eigenvalue. In the case $I = (s,\infty)$,
$\tilde{E}_N(I;a,\mu)$ is simply related to an average in the Jacobi unitary
ensemble, and this in turn is simply related to certain averages over the
orthogonal group, the unitary symplectic group and the circular unitary
ensemble. The latter integrals are of interest for their combinatorial content.
Also considered are the hard edge and soft edge scaled limits of
$\tilde{E}_N(I;a,\mu)$. In particular, in the hard edge scaled limit it is
shown that the limiting quantity $E^{\rm hard}((0,s);a,\mu)$ can be evaluated
as a $\tau$-function associated with the Hamiltonian in Okamoto's theory of the
Painlev\'e III equation.
