# LOTUS for Laplace Space

Let $$f(t)$$ be the density of a non-negative, absolutely continuous random variable $$T$$. While $$f(t)$$ has no closed form representation, suppose its Laplace transform $$\int_0^\infty e^{-st}f(t)\,\text{d}t = F_T(s)$$ does have one, and is continuously differentiable. Given $$z \in \mathbb{R}$$ and $$z = n - \alpha$$ with $$\alpha > 0$$, this potentially allows to calculate moments in closed form via $$$$\label{eq:rkl} \mathbb{E}[T^z] = \frac{(-1)^n}{\Gamma(\alpha)} \int_0^\infty F_T^{(n)}(s) s^{\alpha-1}\,\text{d}s.$$$$ My questions:

1. Suppose $$\phi:\mathbb{R}_+ \to \mathbb{R}_+$$ is smooth, strictly increasing, and strictly concave. Given the knowledge above, what, if anything, can I infer about $$\mathbb{E}[\phi(T^z)]$$? I was hoping for an analog of the LOTUS in Laplace space.
2. Is the case $$\phi = W_0$$ of the Lambert function's principal branch special by any chance?
3. Under which circumstances, if any, does knowledge about $$F_T(s)$$ help to find $$F_{\phi(T)}(s)$$?
• TIL people use LOTUS for the Law of the Unconscious Statistician. Jun 8, 2021 at 0:21
• Thanks for the clarification. What does "TIL" stand for? Jun 8, 2021 at 12:17
• "Today I learned," from the subreddit of the same name. Regarding your question, I don't know if (1) has a dual - i.e., inference about expectation of a well-behaved function. Jun 8, 2021 at 15:50