Often, each outcome of an experiment is characterized by a number. If the
outcome is observed as a physical quantity, the size of that quantity (in prescribed units)is the entity actually observed. In many nonnumerical cases, it is convenient to assign
a number to each outcome. For example, in a coin flipping experiment, a “head” may berepresented by a 1 and a “tail” by a 0. In a Bernoulli trial, a success may be represented
by a 1 and a failure by a 0. In a sequence of trials, we may be interested in the numberof successes in a sequence of n component trials. One could assign a distinct number to
each card in a deck of playing cards. Observations of the result of selecting a cardcould be recorded in terms of individual numbers. In each case, the associated number
becomes a property of the outcome.The fundamental idea of a real random variable is the assignment of a real number to each elementary outcome ω in the basic space Ω. Such an assignment amounts to determining a function X, whose domain is Ω and whose range is a subset of the real line R. Each ω is mapped into exactly one value t, although several ω may have the same image point.
Except in special cases, we cannot write a formula for a random variable X. However, random variables share some important general properties of functions which play an essential role in determining their usefulness.Associated with a function X as a mapping are the inverse mapping and the inverse images it produces. By the inverse image of a set of real numbers M under the mapping X, we mean the set of all those ω∈Ω which are mapped into M by X. If X does not take a value in M, the inverse image is the empty set (impossible event). If M includes the range of X, (the set of all possible values of X), the inverse image is the entire basic space Ω.
The class of inverse images of the Borel sets on the real line play an essential role in probability analysis.
Introduction
Probability associates with an event a number which indicates the likelihood of the
occurrence of that event on any trial. An event is modeled as the set ofthose possible outcomes of an experiment which satisfy a property or proposition
characterizing the event.
Often, each outcome is characterized by a number. The experiment is performed. If the
outcome is observed as a physical quantity, the size of that quantity (in prescribed units)is the entity actually observed. In many nonnumerical cases, it is convenient to assign
a number to each outcome. For example, in a coin flipping experiment, a “head” may berepresented by a 1 and a “tail” by a 0. In a Bernoulli trial, a success may be represented
by a 1 and a failure by a 0. In a sequence of trials, we may be interested in the numberof successes in a sequence of
n component trials. One could assign a distinct number to
each card in a deck of playing cards. Observations of the result of selecting a cardcould be recorded in terms of individual numbers. In each case, the associated number
becomes a property of the outcome.