StatlectThe Digital Textbook
Index > Fundamentals of probability > Expected value

Computing the Riemann-Stieltjes integral: some rules

In the lecture on the Expected value we have discussed a rigorous definition of expected value that involves the Riemann-Stieltjes integral. We present here some rules for computing the Riemann-Stieltjes integral. Since we are interested in the computation of the expected value, we focus here on rules that can be applied when the integrator function is the distribution function of a random variable X, that is, we limit our 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]

Solution

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

The book

Most learning materials found on this website are now available in a traditional textbook format.