A probabilistic view on possibility distributions

In this work we shall give a probabilistic interpretation of possibilistic expected value, variance, covariance and correlation. 1 Probability and possibility In probability theory, the dependency between two random variables can be characterized through their joint probability density function. Namely, if X and Y are two random variables with probability density functions fX(x) and fY (y), respectively, then the density function, fX,Y (x, y), of their joint random variable (X,Y ), should satisfy the following properties ∫ R fX,Y (x, t)dt = fX(x), ∫ R fX,Y (t, y)dt = fY (y) for all x, y ∈ R. Furthermore, fX(x) and fY (y) are called the the marginal probability density functions of random variable (X,Y ). X and Y are said to be independent if the relationship fX,Y (x, y) = fX(x)fY (y) holds for all x, y. The expected value of random variable X is defined as