The vacuum state -- or any other state of finite energy -- is not an eigenstate of any smeared (averaged) local quantum field. The outcomes (spectral values) of repeated measurements of that averaged local quantum field are therefore distributed according to a non-trivial probability distribution. In this paper, we study probability distributions for the smeared stress tensor in two dimensional conformal quantum field theory. We first provide a new general method for this task based on the famous conformal welding problem in complex analysis. Secondly, we extend the known moment generating function method of Fewster, Ford and Roman. Our analysis provides new explicit probability distributions for the smeared stress tensor in the vacuum for various infinite classes of smearing functions. All of these turn out to be given in the end by a shifted Gamma distribution, pointing, perhaps, at a distinguished role of this distribution in the problem at hand.
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