laplace transform 拉普拉斯变换

参考网址：

1. https://en.wikipedia.org/wiki/First-hitting-time_model

2. https://en.wikipedia.org/wiki/Laplace_transform

Probability theory

By abuse of language, this is referred to as the Laplace transform of the random variable X itself. Replacing s by −t gives the moment generating function of X. The Laplace transform has applications throughout probability theory, including first passage times of stochastic processessuch as Markov chains, and renewal theory.

Of particular use is the ability to recover the cumulative distribution function of a continuous random variable X by means of the Laplace transform as follows^[11]

{\displaystyle F_{X}(x)={\mathcal {L}}^{-1}\!\left\{{\frac {1}{s}}E\left[e^{-sX}\right]\right\}\!(x)={\mathcal {L}}^{-1}\!\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}\!(x).}