Editing Laplace transform (section)

=== Probability theory ===
In [[probability theory|pure]] and [[applied probability]], the Laplace transform is defined as an [[expected value]]. If {{mvar|X}} is a [[random variable]] with probability density function {{mvar|f}}, then the Laplace transform of {{mvar|f}} is given by the expectation
<math display=block>
 \mathcal{L}\{f\}(s) = \operatorname{E}\left[e^{-sX}\right],
</math>
where <math>\operatorname{E}[r]</math> is the [[Expected value|expectation]] of [[random variable]] <math>r</math>.

By [[Abuse of notation|convention]], this is referred to as the Laplace transform of the random variable {{mvar|X}} itself. Here, replacing {{mvar|s}} by {{math|−''t''}} gives the [[moment generating function]] of {{mvar|X}}. The Laplace transform has applications throughout probability theory, including [[first passage time]]s of [[stochastic process]]es such as [[Markov chain]]s, and [[renewal theory]].

Of particular use is the ability to recover the [[cumulative distribution function]] of a continuous random variable {{mvar|X}} by means of the Laplace transform as follows:<ref>The cumulative distribution function is the integral of the probability density function.</ref>
<math display=block>
 F_X(x) = \mathcal{L}^{-1}\left\{\frac{1}{s} \operatorname{E}\left[e^{-sX}\right]\right\}(x) = \mathcal{L}^{-1}\left\{\frac{1}{s} \mathcal{L}\{f\}(s)\right\}(x).
</math>