Recovering Fisher-Information from the MGF Alone without Requiring Explicit PMF or PDF from a One-Parameter Exponential Family

It is well-known that a finite moment generating function (m.g.f.) corresponds to a unique probability distribution. So, an important question arises: Is it possible to obtain an expression of Fisher-information, I_X(Ɵ); using the m.g.f. alone, that is without requiring explicitly a probability mass function (p.m.f.) or probability density function (p.d.f.), given that the p.m.f. or p.d.f came from a one-parameter exponential family? We revisit the core of statistical inference by developing a clear link (Theorem 1.1) between the m.g.f. and I_X(Ɵ). Illustrations are included.