  We derive exact expressions for the finite-time statistics of extrema
(maximum and minimum) of the spatial displacement and the fluctuating entropy
flow of biased random walks. Our approach captures key features of extreme
events in molecular motor motion along linear filaments. Our results generalize
the infimum law for entropy production and reveal a symmetry of the
distribution of its maxima and minima. We also show that the relaxation
spectrum of the full generating function, and hence of any moment, of the
finite-time extrema distributions can be written in terms of the
Mar{\v{c}}enko-Pastur distribution of random-matrix theory. Using this result,
we obtain estimates for the extreme-value statistics of stochastic transport
from the eigenvalue distributions of suitable Wishart and Laguerre random
matrices. We confirm our results by numerical simulations of stochastic models
of molecular motors and discuss as illustrative example our theory in the
context of sports.
