Probability distributions of smeared quantum stress tensors

We obtain in closed form the probability distribution for individual measurements of the stress-energy tensor of two-dimensional conformal field theory in the vacuum state, smeared in time against a Gaussian test function. The result is a shifted gamma distribution with the shift given by the previously known optimal quantum inequality bound. For small values of the central charge it is overwhelmingly likely that individual measurements of the sampled energy density in the vacuum give negative results. For the case of a single massless scalar field, the probability of finding a negative value is 84%. We also report on computations for four-dimensional massless scalar fields showing that the probability distribution of the smeared square field is also a shifted gamma distribution, but that the distribution of the energy density is not.