An algorithm framework of the numerical inversion of Laplace transforms(NILT)based on a class of probability density function

which belongs to the sequence of -type functions with exponential term exp(-st)was presented.If the probability density function defined for all positive variables in the abscissa has and only has an extremum abscissa corresponding to the modal value of random variables

the expectation of a continuous function of random variables can be expressed by the Laplace transform of this function.It is shown that the traditional Stehfest's and Post-Widder's NILT algorithm are two special cases of the proposed algorithm framework.Moreover

an example using the

-type Gamma distribution density function i

s given to demonstrate that the novel framework can help to develop new NILT algorithms.