3 Random variables and Probability Distribu tions The model based on(S, F, p) does not provide us with a flexible enough frame- work. The basic idea underlying the construction of (S, F, p) was to set up a framework for studying probability of events as a prelude to analyzing problem involving uncertainty. One facet of E which can help us suggest a more flexible probabilities space is the fact when the experiment is performed the outcome is often considered in relation to some quantifiable attribute; i.e. an attribute which can be repressed in numbers. It turns out that assigning numbers to qual itative outcome make possible a much more flexible formulation of probability theory. This suggests that if we could find a consistent way to assign numbers to outcomes we might be able to change(, F, p)to something more easily handled The concept of a random variable is designed to just that without changing the underlying probabilistic structure of(S, F, p) 3.1 The Concept of a random variable Let us consider the possibility of defining a function X( which maps s directly into the real line R. that is assigning a real number ai to each s1 in S by 1=X(s1, al ER, S1 E S. The question arises as to whether every function from s to R will provided us with a consistent way of attaching numbers to elementary events; consistent in the sense of preserving the event structure of the probability space(S, F, P). The answer, unsurprisingly, is not. This is because, although X is a function defined on S, probabilities are assigned to events in F and the issue we have to face is how to define the value taken by x for the different elements of s in a way which preserve the event structures of F. What we require from X-or (X) is to provide us with a correspondence between Rr and S which reflects the event structure of that is, it preserves union, intersections and complement n other word for each subset N of Rx, the inverse image X-(N) must be an event in F. This prompts us to define a random variable X to be any function satisfying this event preserving condition in relation to some a-field defined on Rr: for generality we always take the borel field B on R3 Random Variables and Probability Distribu￾tions The model based on (S, F,P) does not provide us with a flexible enough frame￾work. The basic idea underlying the construction of (S, F,P) was to set up a framework for studying probability of events as a prelude to analyzing problem involving uncertainty. One facet of E which can help us suggest a more flexible probabilities space is the fact when the experiment is performed the outcome is often considered in relation to some quantifiable attribute; i.e. an attribute which can be repressed in numbers. It turns out that assigning numbers to qual￾itative outcome make possible a much more flexible formulation of probability theory. This suggests that if we could find a consistent way to assign numbers to outcomes we might be able to change (S, F,P) to something more easily handled. The concept of a random variable is designed to just that without changing the underlying probabilistic structure of (S, F,P). 3.1 The Concept of a Random Variable Let us consider the possibility of defining a function X(·) which maps S directly into the real line R, that is, X(·) : S → Rx, assigning a real number x1 to each s1 in S by x1 = X(s1), x1 ∈ R, s1 ∈ S. The question arises as to whether every function from S to Rx will provided us with a consistent way of attaching numbers to elementary events; consistent in the sense of preserving the event structure of the probability space (S, F,P). The answer, unsurprisingly, is not. This is because, although X is a function defined on S, probabilities are assigned to events in F and the issue we have to face is how to define the value taken by X for the different elements of S in a way which preserve the event structures of F. What we require from X −1 (·) or (X) is to provide us with a correspondence between Rx and S which reflects the event structure of F, that is, it preserves union, intersections and complements. In other word for each subset N of Rx, the inverse image X−1 (N) must be an event in F. This prompts us to define a random variable X to be any function satisfying this event preserving condition in relation to some σ-field defined on Rx; for generality we always take the Borel field B on R. 13