Operating on Functions with Variable Domains

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of and, or, not and conditioning are then defined on these conditional events with variable conditions. The probabilities of the disjunction (or) and of the conjunction (and) of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed.