  We study the statistics of charge transfer through an impurity in a chiral
Luttinger liquid (realized experimentally as a quantum point contact in a
fractional quantum Hall edge state device). Taking advantage of the
integrability we present a procedure for obtaining the cumulant generating
function of the probability distribution to transfer a fixed amount of charge
through the constriction. Using this approach we analyze in detail the
behaviour of the third cumulant C_3 as a function of applied voltage,
temperature and barrier height. We predict that C_3 can be used to measure the
fractional charge at temperatures, which are several orders of magnitude higher
than those needed to extract the fractional charge from the measurement of the
second cumulant. Moreover, we identify the component of C_3, which carries the
information about the fractional charge.
