  Brownian motion is a Gaussian process described by the central limit theorem.
However, exponential decays of the positional probability density function
$P(X,t)$ of packets of spreading random walkers, were observed in numerous
situations that include glasses, live cells and bacteria suspensions. We show
that such exponential behavior is generally valid in a large class of problems
of transport in random media. By extending the Large Deviations approach for a
continuous time random walk we uncover a general universal behavior for the
decay of the density. It is found that fluctuations in the number of steps of
the random walker, performed at finite time, lead to exponential decay (with
logarithmic corrections) of $P(X,t)$. This universal behavior holds also for
short times, a fact that makes experimental observations readily achievable.
