  The largest eigenvalue distribution of the Wishart-Laguerre ensemble, indexed
by Dyson parameter $\beta$ and Laguerre parameter $a$, is fundamental in
multivariate statistics and finds applications in diverse areas. Based on a
generalization of the Selberg integral, we provide an effective recursion
scheme to compute this distribution explicitly in both the original model, and
a fixed-trace variant, for $a,\beta$ non-negative integers and finite matrix
size. For $\beta = 2$ this circumvents known symbolic evaluation based on
determinants which become impractical for large dimensions. Our exact results
have immediate applications in the areas of multiple channel communication and
bipartite entanglement. Moreover, we are also led to the exact solution of a
long standing problem of finding a general result for Landauer conductance
distribution in a chaotic mesoscopic cavity with two ideal leads. Thus far,
exact closed-form results for this were available only in the Fourier-Laplace
space or could be obtained on a case-by-case basis.
