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Computing the Riemann-Stieltjes integral: some rules

We present here some rules for computing the Riemann-Stieltjes integral when the integrator function is the distribution function of a random variable X, i.e. we limit attention to integrals of the kind:[eq1]where [eq2] is the distribution function of a random variable X and [eq3]. Before stating the rules, note that the above integral does not necessarily exist or is not necessarily well-defined. Roughly speaking, for the integral to exist the integrand function $g$ must be well-behaved. For example, if $g$ is continuous on $left[ a,b ight] $, then the integral exists and is well-defined.

That said, we are ready to present the calculation rules:

  1. [eq2] is continuously differentiable on $left[ a,b ight] $. If [eq2] is continuously differentiable on $left[ a,b ight] $ and [eq6] is its first derivative, then:[eq7]

  2. [eq2] is continuously differentiable on $left[ a,b ight] $ except at a finite number of points. Suppose [eq9] is continuously differentiable on $left[ a,b ight] $ except at a finite number of points $c_{1}$, ..., $c_{n}$ such that:[eq10]Denote the derivative of [eq2] (where it exists) by [eq12]. Then:[eq13]

Exercise

Let [eq2] be defined as follows:[eq15]where $lambda >0$.

Compute the following integral:[eq16]

nav_button Solution

[eq2] is continuously differentiable on the interval $left[ 1,2 ight] $. Its derivative [eq18] is:[eq19]As a consequence, the integral becomes:[eq20]

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