Statistics for Maintenance Modelling: Renewal Processes

# Laplace Tranforms¤

We define the Laplace transform of f(x) as:

(s) = f(x) dx, where s > 0.

The inverse transform is made either by using a table of transforms of common functions, or by doing a contour integration.

The pdf. of the sum of two random variables is given by the convolution of the two pdfs, and a very useful feature of Laplace transforms is that the transform of a convolution is the product of transforms of the two convolved functions. This means that, on taking transforms, a pdf. that is a k-fold convolution is replaced by the kth power of the Laplace transform of the failure time.

Mathematically, convolution means that if X1 and X2 are random variates, the pdf f(t) of X + X1 + X2 is given by
f(t) = f1(y)f2(t - y) dy, and the Laplace transforms obey (s) =1(s)2(s)

Laplace transforms are therefore very useful in renewal theory, because the series of convolutions of probabilities that crop up become simple geometric series when expressed as Laplace transforms.

You can however follow the essentials of renewal theory without bothering about these transforms, or come back to them later. For a fuller account, and a table of important Laplace transforms see the Laplace Transforms page on the Mathworld website.