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.
Let f(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.