FRVs play an important role in developing multi-objective fuzzy stochastic programming models. To get a brief concept on FRVs, the concept of random variables is highly needed by the readers. From that view point a short discussion on basic probability theory with various types of probability distributions are briefly highlighted at first in this section, so that the readers can easily capture the idea of FRVs and can distinguish the difference between random variables and FRVs.
Definition 2.1.1: (Algebra): Let be a random experiment and be its sample space. A non-empty collection of subsets of is said to form an algebra, if the following conditions are satisfied
if , then .
Definition 2.1.2: (Probability Space): Let be a algebra. A mapping that maps each element of to a real number, is called probability measure. If is a sample space, then the triple is called a probability space.
Definition 2.1.3: (Random Variables): Let be a random experiment and be its sample space. If for each event point of the sample space , there is a real number by a given rule, i.e., if is a mapping from the sample space to the set of real numbers , i.e., , such that for all , then is called a random variable or a stochastic variable.