Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 199Let us note in passing that 2, the coefficient of excess, defined by Equation(4.12), for a normal distribution is zero. Hence, it is used as the referencedistribution for 2.7.2.1 THE CENTRAL LIMIT THEOREMThe great practical importance associated with the normal distributionstems from the powerful central limit theorem stated below (Theorem 7.1).Instead of giving the theorem in its entire generality, it serves ourpurposes quite well by stating a more restricted version attributable toLindberg (1922).Theorem 7.1: the central limit theorem. Let fX n g be a sequence of mutuallyindependent and identically distributed random variables with means m andvariances 2 .LetY ˆ Xnjˆ1X j ;…7:14†and let the normalized random variable Z be defined asZ ˆ …Y nm†n 1=2 : …7:15†Then the probability distribution function of Z,F Z (z), converges to N(0, 1) asn !1for every fixed z.Proof of Theorem 7.1: We first remark that, following our discussion inSection 4.4 on moments of sums of random variables, random variable Ydefined by Equation (7.14) has mean nm and standard deviation n 1/2 . Hence,Z is simply the standardized random variable Y with zero mean and unitstandard deviation. In terms of characteristic functions X(t) of random variablesX j , the characteristic function of Y is simply Y …t† ˆ‰ X …t†Š n :…7:16†Consequently, Z possesses the characteristic function jmt t n Z …t† ˆ exp n 1=2 X : …7:17† n 1=2 TLFeBOOK