Given a density $f(t)$, say over $t\in \mathbb{R}^n$ absolutely continuous with Lebesgue measure. Write $u(t) = \log(f(t))$ taking values in $[-\infty, \infty)$ so $f = \exp(u)$.
Can we find a general formula relating the variance of random variable $T\sim f$ in terms of $u(t)$? If so a special case of this would provide variance formulas for any exponential distribution, e.g., the Laplace, Gaussian, ...
One way you can get something helpful is to take the derivative of the expression. you will get: f(t)= (t^k)exp[u(t)]------df(t)/dt= kt^[k-1]exp[u(t)]+t(^k)+u'(t)exp[u(t)] for every k. you get a series of differential equations. So integrating by part we get relation between mk (moment order k), and m(k+1) (moment order k+1). In case you have explicit expression of u(t). One particular case is when u(t) is a polynomial where if you do the computation of derivative with respect to the coefficients of the polynomial, you have an expression relating moments of higher orders. I'll allow myself to mention my PhD thesis, where you have this case treated: Construction of Random Signals from their Higher Order Moments. (I Chamseddine)
