The Generalized Inverse of Distribution Functions
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The generalized inverse will be introduced for distribution functions of arbitrary real-valued random variables. The generalized inverse exists even if the ordinary inverse does not and they are identical if both exist. Further basic properties of the generalized inverse and connections to order relations are discussed. Also, the generalized inverse can be obtained from a sequence of ordinary inverses of suitable distribution functions; sufficient conditions therefore are stated.
A generalized inversion process for the generalized inverse is given so that the generalized inverse of a generalized inverse distribution function reproduces the distribution function. When expectations are finite, they can be computed from the generalized inverse.