Is it possible to evaluate/simplify the following integral involving the Laplace transform? If yes, what are the steps or hints? $$ I = \int_{0}^{\infty}\left[\,1 - \mathcal{L}_{X}\left(a \over y\right)\right]^{k} y^{b-1}\,\mathrm{d}y, $$ where $\mathcal{L}_{X}\left(\cdot\right)$ is the Laplace transform of a positive random variable $X$. $\quad a > 0,\ b \in \left[0, 1\right],\ $ and $\ k \in \mathbb{N}$.
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$\begingroup$ I encountered this problem while calculating higher moments of a random variable. I have edited the question. I am particularly interested in calculating the Laplace transform $X$, where $X$ is a positive random variable. For example, if $X$ is the exponential random variable, we can obtain a closed-form expression. $\endgroup$– user389066Commented Aug 5, 2017 at 15:55
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$\begingroup$ This is still unclear. $\mathcal{L}_X(s) = \mathbb{E}[e^{-sX}]$. Then what ? What is the concrete problem you want to solve, and how did you come to this integral ? $\endgroup$– reunsCommented Aug 5, 2017 at 21:41
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The Laplace transform of $\dfrac{1}{y}$ fails to exist. Look here for the details
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$\begingroup$ Thanks. I have edited the question. I am particularly interested in calculating the Laplace transform $X$, where $X$ is a positive random variable. For example, if $X$ is the exponential random variable, we can obtain a closed-form expression. $\endgroup$– user389066Commented Aug 5, 2017 at 15:56