Laplace transform and fractional moments. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:44:12Z http://mathoverflow.net/feeds/question/5525 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5525/laplace-transform-and-fractional-moments Laplace transform and fractional moments. Piotr Miłoś 2009-11-14T16:17:58Z 2010-06-15T21:30:53Z <p>Is there any "easy" way to calculate fractional moments from Laplace transform. To be more specyfic let us consider the following example. Let $X$ be a positive random variable and $L(\theta) := E \exp (-\theta X)$ be its Laplace transform. Of course it is easy to calculate $E X^n$ where $n$ is a natural number but what with e.g. $E X^{1/2}$.</p> http://mathoverflow.net/questions/5525/laplace-transform-and-fractional-moments/5527#5527 Answer by David Bar Moshe for Laplace transform and fractional moments. David Bar Moshe 2009-11-14T16:40:37Z 2009-11-14T16:40:37Z <p>For your example of X^{1/2} you can evaluate the fractional half derivative of the Laplace transform (see for example the Wikipedia article on fractional calculus) at theta = 0.</p> http://mathoverflow.net/questions/5525/laplace-transform-and-fractional-moments/28317#28317 Answer by Jeff Schenker for Laplace transform and fractional moments. Jeff Schenker 2010-06-15T21:24:36Z 2010-06-15T21:30:53Z <p>If $X$ is positive the following works. Let $F(\theta)=E(e^{-\theta X})$ be the Laplace transform. Given $s\in \mathbb{R}$ write $s=n-\alpha$ with $n$ an integer and $\alpha >0$. Then $$E(X^{s}) = (-1)^n\frac{1}{\Gamma(\alpha)} \int_0^\infty F^{(n)} (\theta) \theta^{\alpha} d \theta$$ with $\Gamma$ the usual Gamma function, $$\Gamma(\alpha) = \int_0^\infty \theta^{\alpha -1} e^{-\theta} d \theta.$$</p> <p>Indeed by Fubini, $$\int_0^\infty F^{(n)}(\theta) \theta^{\alpha-1} d \theta = (-1)^n E (\int_0^\infty X^n e^{-\theta X} \theta^{\alpha-1} d \theta )= (-1)^n E(X^{n-\alpha}) \int_0^\infty \theta^{\alpha-1} e^{-\theta} d \theta,$$ and so long as $\alpha >0$ the integral on the right is convergent.</p>