Moment (probability) and Laplace transforms

First what is a moment? In statistics, a moment is a specific quantitative measure of the shape of a set of points. Now, here’s how the Laplace transform aid in probability.

The Laplace transform is a holomorphic function. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. This means that its domain includes complex numbers and that there exists a complex derivative for the function. A holomorphic function is infinitely differentiable and equal to its own Taylor series. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory.

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