Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 43One can also give PDF and pmf a useful physical interpretation. In terms ofthe distribution of one unit of mass over the real line 1 < x < 1, the PDF ofa random variable at x, F X x), can be interpreted as the total mass associatedwith point x and all points lying to the left of x. The pmf, in contrast, showsthe distribution of this unit of mass over the real line; it is distributed at discretepoints with the amount of mass equal to p X x i )atx i , i ˆ 1,2,....x iExample 3.2. A discrete distribution arising in a large number of physicalmodels is the binomial distribution. Much more will be said of this importantdistribution in Chapter 6 but, at present, let us use it as an illustration forgraphing the PDF and pmf of a discrete random variable.A discrete random variable X has a binomial distribution whenp X …k† ˆ n p k …1 p† n k ; k ˆ 0; 1; 2; ...; n; …3:8†kwhere n and p are two parameters of the distribution, n being a positive integer,and 0 < p < 1. The binomial coefficientnkis defined bynn!ˆk k!…n k†! : …3:9†The pmf and PDF of X for n ˆ 10 and p ˆ 0:2 are plotted in Figure 3.4.p X (x)F X (x)1.00.40.30.20.10.80.60.40.20 2 4 6 8 10x0 2 4 6 8 10x(a)(b)Figure 3.4 (a) Probability mass function, p X ( x), and (b) probability distributionfunction, F X ( x), for the discrete random variable X described in Example 3.2TLFeBOOK