indikaatorfunktsioon

Indikaatorfunktsioon, also known as the characteristic function or indicator function, is a mathematical function used in set theory and probability theory. It is defined for a given set S and an element x. The value of the indicator function is 1 if the element x belongs to the set S, and 0 if the element x does not belong to the set S. Mathematically, if we denote the indicator function of a set S as I_S(x) or 1_S(x), then:

I_S(x) = 1 if x ∈ S

I_S(x) = 0 if x ∉ S

This function is particularly useful for representing the membership of elements within a set in a numerical way. In probability theory, the indicator function is used to define events. For example, if Ω is the sample space and A is an event, then the indicator function of A, I_A, takes the value 1 if the outcome ω ∈ Ω is in A, and 0 otherwise.

Indicator functions simplify many mathematical operations and proofs. For instance, the expected value of an indicator function of an event A is equal to the probability of that event, E[I_A] = P(A). This property makes indicator functions a fundamental tool in various fields, including statistics, machine learning, and economics, for modeling and analyzing discrete choices and occurrences.