The question which arises is whether, in order to define the set function Px(.) we need to consider all the elements of the borel field B. The answer is that we do not need to do that because, as argued above any such element of B can be expressed in terms of the semi-closed intervals(oo, This implies that by choosing such semi-closed intervals intelligently, we can define Px() with the minimum of effort. For example, we may define 0 X<0. P2(-∞,x]) X≤0( That is x=0) X≤1( That is r=1) X≤2(That we can see. the semi-closed intervals were chosen to divided the real line at the points corresponding to the value taken by X. This way of defining the semi-closed intervals is clearly non-unique but will prove very convenient in the next subsection In fact, the event and probability structure of(S, F, P()) is preserved in the induced probability space(R, B, Pr(). We traded S, a set of arbitrary elements for R, the real line; F a o-field of subset of S with B, the Borel field on the real line; and P() a set function defined on arbitrary sets with Px(), a set function on semi-closed intervals of the real line
The question which arises is whether, in order to define the set function PX(·), we need to consider all the elements of the Borel field B. The answer is that we do not need to do that because, as argued above, any such element of B can be expressed in terms of the semi-closed intervals (−∞, x]. This implies that by choosing such semi-closed intervals ’intelligently’, we can define PX(·) with the minimum of effort. For example, we may define: Px((−∞, x]) = 0 X < 0, 1 4 X ≤ 0 (That is x = 0), 3 4 X ≤ 1 (That is x = 1), 1 X ≤ 2 (That is x = 2), As we can see, the semi-closed intervals were chosen to divided the real line at the points corresponding to the value taken by X. This way of defining the semi-closed intervals is clearly non-unique but will prove very convenient in the next subsection. In fact, the event and probability structure of (S, F,P(·)) is preserved in the induced probability space (R, B, Px(·)). We traded S, a set of arbitrary elements, for R, the real line; F a σ-field of subset of S with B, the Borel field on the real line; and P(·) a set function defined on arbitrary sets with PX(·), a set function on semi-closed intervals of the real line. 16
3.2 The Distribution and Density Functions In the previous section the introduction of the concept of a random variable X enable us to trade the probability space(S, F, P() for(R, B, Px()) which has a much more convenient mathematical structure. The latter probability space however, is not as yet simple enough because Px() is still a set function albeit on real line intervals. In order to simplify it we need to transform it into a point function with which we are so familiar Define a point function F():R→[0,1], which is seemingly, only a function of x. In fact, however, this function will do exactly the same job as Px(). Heuristically, this is achieved by defining F(as a point function by Px(-∞,x])=F(x)-F(-∞), for all aT∈R, and assigning the value zero to F(oo) Definition 8 Let X be a r.v. defined on(S, F, P(). The point function F(: R-0,1 F(x)=P(-∞,x)=Pr(X≤x), for all a∈R is called the distribution function(DF) of X and satisfied the following prop- erties: (a). F() is non-decreasing; (b).F(-∞)=limx→-∞F(x)=0andF(∞)=limn→∞F(x)=1, (c). F(a) is continuous from the right. (i.e. limh-o F(r+h)=F(), Va E R The great advantage of F( over P( and Px()is that the former is a point function and can be represented in the form of an algebraic formula; the kind of functions we are so familiar with in elementary mathematics Definition 9: A random variable X is called discrete if its range r is some subsets of the set
3.2 The Distribution and Density Functions In the previous section the introduction of the concept of a random variable X, enable us to trade the probability space (S, F,P(·)) for (R, B, PX(·)) which has a much more convenient mathematical structure. The latter probability space, however, is not as yet simple enough because PX(·) is still a set function albeit on real line intervals. In order to simplify it we need to transform it into a point function with which we are so familiar. Define a point function F(·) : R → [0, 1], which is seemingly, only a function of x. In fact, however, this function will do exactly the same job as PX(·). Heuristically, this is achieved by defining F(·) as a point function by PX((−∞, x]) = F(x) − F(−∞), for all x ∈ R, and assigning the value zero to F(−∞). Definition 8: Let X be a r.v. defined on (S, F,P(·)). The point function F(·) : R → [0, 1] defined by F(x) = Px((−∞, x]) = Pr(X ≤ x), for all x ∈ R is called the distribution function (DF) of X and satisfied the following properties: (a). F(x) is non-decreasing; (b). F(−∞)=limx→−∞F(x) = 0 and F(∞)=limx→∞F(x) = 1, (c). F(x) is continuous from the right. (i.e. limh→0F(x + h) = F(x), ∀x ∈ R.) The great advantage of F(·) over P(·) and PX(·) is that the former is a point function and can be represented in the form of an algebraic formula; the kind of functions we are so familiar with in elementary mathematics. Definition 9: A random variable X is called discrete if its range Rx is some subsets of the set 17