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 set3.2 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 prop￾erties: (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