We present a weakest-precondition-style calculus for reasoning about the expected values (pre-expectations) of emph{mixed-sign unbounded} random variables after execution of a probabilistic program. The semantics of a while-loop is well-defined as the limit of iteratively applying a functional to a zero-element just as in the traditional weakest pre-expectation calculus, even though a standard least fixed point argument is not applicable in this context. A striking feature of our semantics is that it is always well-defined, even if the expected values do not exist. We show that the calculus is sound, allows for compositional reasoning, and present an invariant-based approach for reasoning about pre-expectations of loops.
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