We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles. In the case $β=1$ this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for $β$ a positive integer, and the parameter $λ_1$ 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 $β= 1$ and $λ_1 + 1/2$ 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 $λ_1$ 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.