More About Expectation

In this section we define the expectation of a function with respect to (i) a probability law specified by its distribution function, and (ii) a numerical -tuple valued random phenomenon.

Stieltjes Integral . In section 2 we defined the expectation of a continuous function with respect to a probability law, which is specified by a probability mass function or by a probability density function. We now consider the case of a general probability law, which is specified by its distribution function .

In order to define the expectation with respect to a probability law specified by a distribution function , we require a generalization of the notion of integral, which goes under the name of the Stieltjes integral . Given a continuous function , a distribution function , and a half open interval on the real line (that is, consists of all the points strictly greater than and less than or equal to ), we define the Stieltjes integral of , with respect to over , written , as follows. We start with a partition of the interval into subintervals , in which are points chosen so that . We then choose a set points , one in each subinterval, so that for , We define

in which the limit is taken over all partitions of the interval , as the maximum length of subinterval in the partition tends to 0.

It may be shown that if is specified by a probability density function , then

whereas if is specified by a probability mass function then

The Stieltjes integral of the continuous function , with respect to the distribution function over the whole real line, is defined by

The discussion in section 2 in regard to the existence and finiteness of integrals over the real line applies also to Stieltjes integrals. We say that exists if and only if is finite. Thus only absolutely convergent Stieltjes integrals are to be invested with sense.

We now define the expectation of a continuous function , with respect to a probability law specified by a distribution function , as the Stieltjes integral of , with respect to over the infinite real line; in symbols,

Stieltjes integrals are only of theoretical interest. They provide a compact way of defining, and working with, the properties of expectation. In practice, one evaluates a Stieltjes integral by breaking it up into a sum of an ordinary integral and an ordinary summation by means of the following theorem: if there exists a probability density function , a probability mass function , and constants and , whose sum is 1, such that for every

then for any continuous function

In giving the proofs of various propositions about probability laws we most often confine ourselves to the case in which the probability law is specified by a probability density function, for here we may employ only ordinary integrals. However, the properties of Stieltjes integrals are very much the same as those of ordinary Riemann integrals; consequently, the proofs we give are immediately translatable into proofs of the general case that require the use of Stièltjes integrals.

Expectations with Respect to Numerical -Tuple Valued Random Phenomena . The foregoing ideas extend immediately to a numerical -tuple valued random phenomenon. Given the distribution function of such a random phenomenon and any continuous function of real variables, we define the expectation of the function with respect to the random phenomenon by

in which the integral is a Stieltjes integral over the space of all -tuples of real numbers. We shall not write out here the definition of this integral.

We note that (6.2) and (6.3) generalize. If the distribution function is specified by a probability density function so that of Chapter 4 holds, then

If the distribution function is specified by a probability mass function , so that of Chapter 4 holds, then

Exercises