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 , that is, we limit our attention to integrals of the kindwhere is the distribution function of a random variable and . 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 must be well-behaved. For example, if is continuous on , then the integral exists and is well-defined.
That said, we are ready to present the calculation rules:
is continuously differentiable on . If is continuously differentiable on and is its first derivative, then
is continuously differentiable on except at a finite number of points. Suppose is continuously differentiable on except at a finite number of points , ..., such thatDenote the derivative of (where it exists) by . Then,
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Let be defined as follows:where .
Compute the following integral:
is continuously differentiable on the interval . Its derivative isAs a consequence, the integral becomes
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