Random variables and probabilities  (Page 8/8)

This solution is not as convenient to write out. However, with the distribution for X as defined, a great many other probabilities can be determined. This is particularly the case when it is desired to compare the results of two independent races or “heats.”We consider such problems in the study of Independent Classes of Random Variables .

A function form for canonic

One disadvantage of the procedure canonic is that it always names the output X and $PX$ . While these can easily be renamed, frequently it is desirable to use some other name for the random variable from the start. A function form, which we call canonicf , is useful in this case.

General random variables

The distribution for a simple random variable is easily visualized as point mass concentrations at the various values in the range, and the classof events determined by a simple random variable is described in terms of the partition generated by X (i.e., the class of those events of the form $A_i=\{X=t_i\}$ for each t _i in the range). The situation is conceptually the same for the general case, but the details are more complicated. If the random variable takes ona continuum of values, then the probability mass distribution may be spread smoothly on the line. Or, the distribution may be a mixture of point mass concentrations andsmooth distributions on some intervals. The class of events determined by X is the set of all inverse images $X^{-1}\left(M\right)$ for M any member of a general class of subsets of subsets of the real line known in the mathematical literature as the Borel sets . There are technical mathematical reasons for not saying M is any subset, but the class of Borel sets is general enough to include any set likely to be encounteredin applications—certainly at the level of this treatment. The Borel sets include any interval and any set that can be formed by complements, countable unions, and countableintersections of Borel sets. This is a type of class known as a sigma algebra of events. Because of the preservation of set operations by the inverse image, the class of eventsdetermined by random variable X is also a sigma algebra, and is often designated $\sigma \left(X\right)$ . There are some technical questions concerning the probability measure P _X induced by X , hence the distribution. These also are settled in such a manner that there is no need for concern at this level of analysis. However, some of these questions becomeimportant in dealing with random processes and other advanced notions increasingly used in applications. Two facts provide the freedom we need to proceed with little concernfor the technical details.

1. $X^{-1}\left(M\right)$ is an event for every Borel set M iff for every semi-infinite interval $(-\infty ,t]$ on the real line $X^{-1}\left((-\infty ,t]\right)$ is an event.
2. The induced probability distribution is determined uniquely by its assignment to all intervals of the form $(-\infty ,t]$ .

These facts point to the importance of the distribution function introduced in the next chapter.

Another fact, alluded to above and discussed in some detail in the next chapter, is that any general random variable can be approximated as closely as pleased by a simple randomvariable. We turn in the next chapter to a description of certain commonly encountered probability distributions and ways to describe them analytically.