Random variables and probabilities  (Page 3/8)

Before considering further examples, we note a general property of inverse images. We state it in terms of a random variable, which maps Ω to the real line (see Figure 3).

Preservation of set operations

Let X be a mapping from Ω to the real line R . If $M,M_i,i\in J$ , are sets of real numbers, with respective inverse images $E,E_i$ , then

Examination of simple graphical examples exhibits the plausibility of these patterns. Formal proofs amount to careful reading of the notation. Central to the structure are the facts thateach element ω is mapped into only one image point t and that the inverse image of M is the set of all those ω which are mapped into image points in M .

An easy, but important, consequence of the general patterns is that the inverse images of disjoint $M,N$ are also disjoint. This implies that the inverse of a disjoint union of M _i is a disjoint union of the separate inverse images.

Mass transfer and induced probability distribution

Because of the abstract nature of the basic space and the class of events, we are limited in the kinds of calculations that can be performed meaningfully with the probabilitieson the basic space. We represent probability as mass distributed on the basic space and visualize this with the aid of general Venn diagrams and mintermmaps. We now think of the mapping from Ω to R as a producing a point-by-point transfer of the probability mass to the real line. This may be done as follows:

To any set M on the real line assign probability mass $P_X\left(M\right)=P\left(X^{-1}\left(M\right)\right)$

It is apparent that $P_X\left(M\right)\ge 0$ and $P_X\left(\mathbf{R}\right)=P\left(\Omega \right)=1$ . And because of the preservation of set operations by the inverse mapping

This means that P _X has the properties of a probability measure defined on the subsets of the real line. Some results of measure theory show that thisprobability is defined uniquely on a class of subsets of R that includes any set normally encountered in applications. We have achieved a point-by-point transfer of theprobability apparatus to the real line in such a manner that we can make calculations about the random variable X . We call P _X the probability measure induced by X . Its importance lies in the fact that $P(X\in M)=P_X\left(M\right)$ . Thus, to determine the likelihood that random quantity X will take on a value in set M , we determine how much induced probability mass is in the set M . This transfer produces what is called the probability distribution for X . In the chapter "Distribution and Density Functions" , we consider useful ways to describe the probability distribution induced by a random variable. We turn first to a special class of random variables.