Accurate and fast approximations of moment - generating functions and their inversion for log - normal and similar distributions ∗

A general approximation formula for the moment-generating function of random variables with broadly distributed logarithms is derived using a saddle-point expansion of the defining integral. As a special case, a simple and accurate approximation formula for the momentgenerating function of log-normal distributions is obtained. For 4.3 dB spread it reaches 0.5% accuracy and improves rapidly as the spread increases. Exact expressions are derived to obtain, from moment-generating functions of log-normals and related random variables, the moments of their logarithms and the expectation values of their rth powers (−∞ < r < 1). Furthermore, an accurate inversion formula that estimates a log-normal-like distribution from the moment generating function and its derivatives is presented. It is conjectured that the formula converges to the exact distribution as higher derivatives are included.