Lecture 5/ Random Variable Part 2 2017 2018 /

Random Variable
For a given probability space (?,?,P[ .]), a random variable, denoted by X or X(•), is a function with domain and counterdomain the real line. The function X( .) must be such that the set Ar , defined by Ar = {w: X(w) < r}, belongs to ? for-every real number r.
If one thinks in terms of a random experiment, n is the totality of out-
comes of that random experiment, and the function, or random variable, X( .) with domain n makes some real n umber correspond to each outcome of the experiment. That is the important part of our definition. The fact that we also require the collection of w s for which X(w) < r to be an event (i.e., an element of d) tor each real number r is not much of a restriction for our purposes since our intention is to use the notion of random variable only in describing events. We will seldom be interested in a random variable per se; rather we will be interested in events defined in terms of random variables. One might note that the P[ .] of our probability space (?,?,P[ .]), is not used in our definition.
The expression "random variable" is a misnomer that has gained such widespread use that it would be foolish for us to try to rename it.
We will try to use capital Latin letters with or without affixes from near the end of the alphabet to denote random variables.
Also, we use the corresponding small letter to denote a value of the random variable.
EXAMPLE 1
Consider the experiment of tossing a single coin. Let the random variable X denote the number of heads. n = {head,tail}, and X(w) = 1 if w = head, and X(w) = 0 if w = tail; so, the random variable X associates a real number with each outcome of the experiment.