We investigate the statistics of microcavity polariton Bose-Einstein condensation by measuring photoluminescence dynamics from a GaAs microcavity excited by single laser excitation pulses. We directly observe fluctuations (jitter) of the polariton condensation onset time and model them using a master equation for the occupancy probabilities. The jitter of the condensation onset time is an inherent property of the condensate formation and its magnitude is approximately equal to the rise time of the condensate density. We investigate temporal correlations between the emission of condensate in opposite circular or linear polarizations by measuring the second-order correlation function $g^{(2)}(t_1,t_2)$. Polariton condensation is accompanied by spontaneous symmetry breaking revealed by the occurrence of random (i.e., varying from pulse to pulse) circular and linear polarizations of the condensate emission. The degree of circular polarization generally changes its sign in the course of condensate decay, in contrast to the degree of linear polarization.
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