  We consider the problem of the exact computation of the marginal eigenvalue
distributions in the Laguerre and Jacobi $\beta$ ensembles. In the case
$\beta=1$ this is a question of long standing in the mathematical statistics
literature. A recursive procedure to accomplish this task is given for $\beta$
a positive integer, and the parameter $\lambda_1$ a non-negative integer. This
case is special due to a finite basis of elementary functions, with
coefficients which are polynomials. In the Laguerre case with $\beta = 1$ and
$\lambda_1 + 1/2$ a non-negative integer some evidence is given of their again
being a finite basis, now consisting of elementary functions and the error
function multiplied by elementary functions. Moreover, from this the
corresponding distributions in the fixed trace case permit a finite basis of
power functions, as also for $\lambda_1$ a non-negative integer. The fixed
trace case in this setting is relevant to quantum information theory and
quantum transport problem, allowing particularly the exact determination of
Landauer conductance distributions in a previously intractable parameter
regime. Our findings also aid in analyzing zeros of the generating function for
specific gap probabilities, supporting the validity of an associated large $N$
local central limit theorem.
