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Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 6.5 The Method of Moment-Generating Functions 319 Solution We have seen in Example 4.16 that Y − µ has moment-generating function e t 2 σ 2 /2 . Hence, ( ) t m Z (t) = E(e tZ ) = E[e (t/σ )(Y −µ) ] = m (Y −µ) = e (t/σ )2 (σ 2 /2) = e t 2 /2 . σ On comparing m Z (t) with the moment-generating function of a normal random variable, we see that Z must be normally distributed with E(Z) = 0 and V (Z) = 1. EXAMPLE 6.11 Solution Let Z be a normally distributed random variable with mean 0 and variance 1. Use the method of moment-generating functions to find the probability distribution of Z 2 . The moment-generating function for Z 2 is m Z 2(t) = E(e tZ2 ) = = ∫ ∞ −∞ ∫ ∞ −∞ e tz2 f (z) dz = ∫ ∞ −∞ 1 √ 2π e −(z2 /2)(1−2t) dz. e tz2 e−z2 /2 √ 2π dz This integral can be evaluated either by consulting a table of integrals or by noting that, if 1 − 2t > 0 (equivalently, t < 1/2), the integrand [ ( z 2 ) ] [ ( z 2 ] exp − (1 − 2t) exp − )/(1 − 2t) −1 2 2 √ = √ 2π 2π is proportional to the density function of a normally distributed random variable with mean 0 and variance (1 − 2t) −1 . To make the integrand a normal density function (so that the definite integral is equal to 1), multiply the numerator and denominator by the standard deviation, (1 − 2t) −1/2 . Then ∫ 1 ∞ ( 1 z 2 ] m Z 2(t) = √ [− )/(1 (1 − 2t) exp − 2t) −1 dz. 1/2 −∞ 2π(1 − 2t) −1/2 2 Because the integral equals 1, if t < 1/2, 1 m Z 2(t) = (1 − 2t) = (1 − 1/2 2t)−1/2 . A comparison of m Z 2(t) with the moment-generating functions in Appendix 2 shows that m Z 2(t) is identical to the moment-generating function for the gammadistributed random variable with α = 1/2 and β = 2. Thus, using Definition 4.10, Z 2 has a χ 2 distribution with ν = 1 degree of freedom. It follows that the density function for U = Z 2 is given by ⎧ ⎨ u −1/2 e −u/2 , u ≥ 0, f U (u) = Ɣ(1/2)21/2 ⎩ 0, elsewhere.

320 Chapter 6 Functions of Random Variables www.downloadslide.com The method of moment-generating functions is often very useful for finding the distributions of sums of independent random variables. THEOREM 6.2 Let Y 1 , Y 2 ,...,Y n be independent random variables with momentgenerating functions m Y1 (t), m Y2 (t),...,m Yn (t), respectively. If U = Y 1 + Y 2 +···+Y n , then m U (t) = m Y1 (t) × m Y2 (t) ×···×m Yn (t). Proof We know that, because the random variables Y 1 , Y 2 ,...,Y n are independent (see Theorem 5.9), m U (t) = E [ e t(Y 1+···+Y n ) ] = E ( e tY 1 e tY2 ···e tY ) n = E ( e tY ) 1 × E ( e tY ) 2 ×···× E ( e tY ) n . Thus, by the definition of moment-generating functions, m U (t) = m Y1 (t) × m Y2 (t) ×···×m Yn (t). EXAMPLE 6.12 Solution The number of customer arrivals at a checkout counter in a given interval of time possesses approximately a Poisson probability distribution (see Section 3.8). If Y 1 denotes the time until the first arrival, Y 2 denotes the time between the first and second arrival, ..., and Y n denotes the time between the (n − 1)st and nth arrival, then it can be shown that Y 1 , Y 2 ,...,Y n are independent random variables, with the density function for Y i given by ⎧ ⎨ f Yi (y i ) = ⎩ 1 θ e−y i /θ , y i > 0, 0, otherwise. [Because the Y i , for i = 1, 2,...,n, are exponentially distributed, it follows that E(Y i ) = θ; that is, θ is the average time between arrivals.] Find the probability density function for the waiting time from the opening of the counter until the nth customer arrives. (If Y 1 , Y 2 ,... denote successive interarrival times, we want the density function of U = Y 1 + Y 2 +···+Y n .) To use Theorem 6.2, we must first know m Yi (t), i = 1, 2,...,n. Because each of the Y i ’s is exponentially distributed with mean θ, m Yi (t) = (1 − θt) −1 and, by Theorem 6.2, m U (t) = m Y1 (t) × m Y1 (t) ×···×m Yn (t) = (1 − θt) −1 × (1 − θt) −1 ×···×(1 − θt) −1 = (1 − θt) −n . This is the moment-generating function of a gamma-distributed random variable with α = n and β = θ. Theorem 6.1 implies that U actually has this gamma distribution and therefore that ⎧ ⎨ 1 f U (u) = Ɣ(n)θ ⎩ n (un−1 e −u/θ ), u > 0, 0, elsewhere.

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