  In one dimensional transport problems the scattering matrix $S$ is decomposed
into a block structure corresponding to reflection and transmission matrices at
the two ends. For $S$ a random unitary matrix, the singular value probability
distribution function of these blocks is calculated. The same is done when $S$
is constrained to be symmetric, or to be self dual quaternion real, or when $S$
has real elements, or has real quaternion elements. Three methods are used:
metric forms; a variant of the Ingham-Seigel matrix integral; and a theorem
specifying the Jacobi random matrix ensemble in terms of Wishart distributed
matrices.
