Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 221The pdf f X (x) in Equation (7.67) is plotted in Figure 7.12 for several valuesof n. It is shown that, as n increases, the shape of f X (x) becomes moresymmetric. In view of Equation (7.68), since X can be expressed as a sum ofidentically distributed random variables, we expect that the 2 distributionapproaches a normal distribution as n !1on the basis of the central limittheorem.The mean and variance of random variable X having a 2 distribution areeasily obtained from Equation (7.57) asm X ˆ n; 2 X ˆ 2n: …7:69†7.5 BETA AND RELATED DISTRIBUTIONSWhereas the lognormal and gamma distributions provide a diversity of onesidedprobability distributions, the beta distribution is rich in providing variedprobability distributions over a finite interval. The beta distribution is characterizedby the density function8>< … ‡ †f X …x† ˆ …† …† x 1 …1 x† 1 ; for 0 x 1;>:0; elsewhere;…7:70†where parameters and take only positive values. The coefficient of f X (x), ‡ )=‰ ) )Š; can be represented by 1/[B, )], whereB…; † ˆ…† …†…‡† ;…7:71†is known as the beta function, hence the name for the distribution given byEquation (7.70).The parameters and are both shape parameters; different combinationsof their values permit the density function to take on a wide variety of shapes.When , >1, the distribution is unimodal, with its peak at x ˆ 1)/ ‡ 2). It becomes U-shaped when ,