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).

F(s) = int_0^infty e^{-st} f(t) ,dt

Then the nth moment of f,

m_n = int_0^infty t^n , f(t), dt

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

(-1)^n F^{(n)}(0) = m_n

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.

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

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