Moments with Laplace

This is a quick note to mention a connection between two recent posts, namely today's post about moments and post from a few days ago about the Laplace transform.

Letf(t) be a function on [0,) and F(s) be the Laplace transform of f(t).

Then the nth moment of f,

is equal to then nth derivative of F, evaluated at 0, with an alternating sign:

To see this, differentiate with respect to s inside the integral defining the Laplace transform. Each time you differentiate you pick up a factor of -t, so differentiating n times you pick up a term (-1)ⁿ tⁿ, and evaluating at s = 0 makes the exponential term go away.

Related posts

• Brief outline of Laplace transforms (pdf)
• Normal approximation to Laplace distribution?
• Computing moments with a fold

The post Moments with Laplace first appeared on John D. Cook.