QUASI-NORMAL DISTRIBUTIONS AND EXPANSION AT THE MODE*

Transcription

1 SLAC-PUB-1233 (9 April 1973 QUASI-NORMAL DISTRIBUTIONS AND EXPANSION AT THE MODE* Yukio Tomozawa Stanford Linear Aelerator Center, Stanford University, Stanford, California Rall Laboratory of Physis The University of Mihigan, Ann Arbor, Mihigan t ABSTRACT The Gram-Charlier series of type A is disussed in terms of deviants whih are related to moments in a way similar to the Hermite polynomials being related to the powers. Distribution funtions are also expressed in terms of the mode moments (umulants or deviants), whih are useful expansions when the distributions are approximately normal. It is shown that suh expan- sions as well as the Gram-Charlier series are valid asymptotially for disrete distributions defined on the semi-infinite interval [ 0, +XJ ]. (Submitted to Communiations in Mathematial Physis. ) * Work supported in part by the U. S. Atomi Energy Commission. t Permanent address.

2 -2-1. Introdution It is an interesting mathematial problem to express distribution funtions in terms of various moments or umulants. The Gram-Charlier (GC) series of type A is one of the solutions whih is useful if the distribution is approximately 2 93 normal (Gaussian). One of the harateristis of suh a distribution - whih we may all a quasi-normal distribution - is that it has a unique maximum, the so-alled mode. In this artile we will show that knowledge of the mode enables us to derive a useful expansion for the distribution funtion. The GC series the expansion around the mode are obtained for a on- tinuous distribution in the range [- ~0, a]. In physis problems, however, we often enounter disrete distributions whih are defined in the range [0, a]. For example, the ross setions an for produing extra n partiles in high energy ollision are defined for the multipliity n = 0, 1, A harateristi feature of the experimental data on multipliity distributions is that they are quasi- norma14-8 the mode the width beo.me larger as the energy inreases. It appears, therefore, that the expansions whih we mentioned earlier are useful for these problems, at lease in an asymptoti sense. We shall show that this is indeed the ase shall desribe the ondition for the validity of the asymptoti expansions. In Setion II, the GC series for distributions defined in the range [-00, w] is disussed by introduing deviants whih are funtions of moments or umulants. It is pointed out that the relationship between the deviants the moments is similar to that between the Hermite polynomials the powers. Setion III deals with expansions at the mode, Setion IV deals with the problem of disrete distributions in the semi-infinite range.

3 Deviants the Gram-Charlier Series Moments,uk umulants ~~ for a distribution funtion f(x) normalized in the range - 00 < x < ~0 are defined through the harateristi funtion (. f. ) where G(t) = / eitx f(x) dx, -o = O I*ktitjk O" Kk(i# = exp k! k=o k=l k! m tik( ) titlk k!, k=2 1, (2.1) (2.2) (2.3) pk(a) = Or-a)k = J- lx - ajk f(x) dx -o (2.4) sts for the klth moment around a point a, k = lyp) l (2.5) Cumulants are related to moments: (2.6) K4 = IL4(% - 3[P2&)12, et.

4 -4- It is onvenient, for our purpose, to introdue deviants hk, o( ~3), by W iklt+$[l + g3 A$$], = e (2.7) (2.8) where? = $K2t (2.9) k = hk/k2k/2. (2.10) Sine the. f. orresponds WI = e iklt +- to the normal distribution K2 (it)2 2 (2.11) 3 o K2 (it)2 (X- K1 ) f(x) 1 iklt + 2 e- itxdt = J21 e- 2K2, = 5 J -o TK 2 (2.12) deviants hk or hk give the measure of deviation from the normal distribution (as do the umulants K~, k L 3). Deviants are related to moments umulants in the following way: - A3 = K3 = P3(X) (2.13) 1. This name was suggested to me by G. West.

5 -5- k = k! lk12j (-l)pk;pk _ 2Q(z, Q=O 2QQ! (k-28)! kz3, (2.14) k! Kk Kk Kkr Kk2 Kk3 = Kk+z k.,3 k;k: +!$ kiz 3 kl!k2!k3! + f2 15) kl+k2=k kl +k2+k3 =k kz3. Using Eq. (10) the similar notation (;;k = \ / K: ~, et. ), we may rewrite Eqs. (14) (15) as hk=k! p&l Q =0 (- 1fi$-2Qtz) 2QQ! (k-28)?, kz3 (2.16) =; k! k+2? Kkl Kk2 k: > 3 kl!k2 +- k! 3! k;r 3 Kkl kz k3 + kl!k2!k3! -. (2.17), kl + k2 = k kl + k2 + k3 = k ks3. We notie that in Eqs. (14) (16), we have the identity /.L1 (2) = z1 (Z) = 0 (2.18) by definition, that the series in Eqs. (15) (17) terminates with the [k/3] - th sum. The inverse of Eq. (16) is given by WI h Fk(x) = k! k- 28 Q=O 2 Q! (k-2q)! (2.19)

6 -6- with the onstraints (2.20) With these preparations, we express the distribution funtion f(x) in terms of deviants : Using Eqs. (l), (8) (9), defining z=$, X-K (2.21) we obtain -itx e q(t) dt (2.22) (2.23) where the identity of the Hermite polynomial Z2 -?!- Z2 -- = (-l)k Hk(z) e 2 (2.24) has been used. Eq. (23) is the Cram-Charlier series of type A. We point out that the reiproal relation between moments deviants, Eqs. (16) (19) resembles that between the powers the Hermite polynomials Q,k-XQ Hk(z) = k! (;I) Q =0 2 I! (k-2q)!

7 -7- zk = k! [!dd Hk-2P) Q = 0 24! (k-21)! (2.26) whih follows from the generating funtion o tjh.(z) e = -. j=o j! (2.27) The only differene is that in the former the first few terms of moments deviants are missing by definition (Eqs. (8) (20)). Using Eqs. (25), we reast the CG series into the form where -- f(x) = -$.g e, z2 * 2 G Zk k - 2 k=() k! 4 6 hg = 1+4 vo !! -... h3 h5 h7 Vl = --F +qy-gtj +*.* 2 = -=+4!! - -(3!! +... v3 = x3 - h5 rr I h7.. 4!! - -** (2.28) (2.29) k = kz3. (2.30)

8 -8- Finally we note the further -- z2 k 2.k z e relations Cl k=o z = i- -.k - k 1 Hk(t) k! z2 (2.31) (2.32) whih are reiproal to Eqs. (8) (24) respetively. Eq. (31) an be obtained from Eqs. (8) (26), Eq. (32) is the Fourier transform of Eq. (24). From Eqs. (22), (31) (32), it follows that f(x) = k=o -- 2 =&k;o~e 2 ~0 - vz k -zk 2 (2.33), whih is idential to Eq. (28). III. Mode Quasi-normal Expansion The mode m is the stationary point of distribution funtions is deter- mined as the solution of the equation 03 - o k=o k ;r zk k+l k k! - k=o k! = z1 -I- (;r 2-vo)z+p - -z1jz (gl - &jzk+... = 0.

9 -9- This an be solved only if a few terms in the series are important, or if Eq. (1) is summable in a ompat form. The former ase was disussed in referenes l-3, 9. In this artile, instead, we onsider the ase where the mode is known. This simplifies the problem enormously as far as a formal manipulation is onerned, as will be seen below. In terms of the mode, we antiipate the expansion 1 f(x) = 4-m exp [-*](,+S3$(gki; (3.2a) (3.2b) These expansion formulae were disussed previously3 for the Poisson distribution temperate borrelation models whih are haraterized by the ondition \ where Kk = O(E k-2 ), kz 3, E << 1. (3.3) (3.4) In the latter ase, we an prove that3 a3 = O(E), = O(E2), a4,6 Q bk = o(e k-2 ), k 2 3, (3.5) (3.6)

10 -lo- thus the oeffiients of higher powers in (x-m)/y are suessively smaller. The distribution learly exhibits a quasi-normal behavior. The mode the width are also alulated 2,3 as expansions in e (3.7) (3 4 The aim of this setion is to derive expansion formula (2) in a more general ase. Assuming that the mode m is already known, the width may be omputed by the formula 1 a2!nf(x) = 1 7=- ax2 T- 2 x=m I +z;- k = O k! GO- k kzm k=o k! \ (3.9) where Z m =^. m - K, 4 (3.10) In order to ompute the other parameters in the expansion formula (2), it would be more onvenient to use the expansion of the. f.

11 -ll- imt 4 (W? P(t) = e 2 [l + g1 yyk] (3.11) = e (3.12) where 5, = k! CJd21 t-l) Q 28 Y b&2qfrn) Q=O ZQQ! (k-28)! (3.13) [k/21 tk = k! lx- t-l)% k-2q(m) Q=o 24! (k-as)! (3.14) A t = yt, (3.15) i, = tk/ yk, ;k k = pk/y, et. (3.16) P,(m) = 5, = 1. (3.17) It is obvious that we have the relation reiproal to Eq. (14) whih is similar to Eq. (2.26), ik(m) = k! k-2q Q =0 2 Q! (k-2q)! * (3.18)

12 -12- is given by An alternative expression of the. f., whih is analogous to Eq. (2.31), k=o 8, ik Hk (t) k! (3.19) where the identity (2.26) was used, $, is related to tk by 03 $k = (- &Q+k, Q=. PQ)!! krl. (3.20) (3.21) As was done previously, the distribution funtion is obtained as the Fourier trans- form of Eq. (19), f(x) = 2 k=o = 1 &y GkikHk(i-$-] 1 - $ k! - e GY o $ yk -- y2 k 2 k!e > k=o (3.22) where (3.23) From the fat that m is the mode y is the width, it follows that A n1 = G2 = 0 (3.24)

13 Eq. (22) an be written in the form ( 2a) with the parameter (3.25) (3.26) The oeffiients bk in Eq. (2b) an be expressed in terms of ak vie versa: kl+k2=k kl+k2+k3=k aklak2ak3 kl!k2! k3? -... (3.27) k = bk+$ b b klk2 k k;3 kl!k2! + % kiz 3 k,:,rk kl+k2+k3=k bklbk2bk3 + kl!k2!k3! * (3.28) The series in Eqs. (27) (28) terminates with the -th sum as in Eq. (2.17). This ompletes the formal derivation of the quasi-normal expansion, Eq. (2). Needless to say, suh expansions are most effetive if ak or bk derease very quikly as k - 00.

14 -14- IV. Disrete Distribution in the Semi-Infinite Range2 For a disrete distribution Pn whih is normalized by 00 n = 1 ) n=o we proeed in a way similar to the preeding setions. The. f. is defined by (4.1) (4.2) most of the formulae onerning the moments, umulants, deviants, the like, are valid also in this ase exept that the integral in x is replaed by the sum over n. For example, the moments are given by Pk(a) = (n- a)k = n=o 00 k (n-a) P. n \ (4.3) Using Eqs. (2.8)-(2.10) (2.31), we invert Eq. (2) to obtain 2. The basi argument in this setion is the same as in referene 3. We present it for ompleteness to inlude the ase of the general expansion formula disussed in the previous setions.

15 -15- N y2 izt-2 d: (4.4) T2 -izx- _ 2 e d?, (4.5) where Z (4.6) The integral in these equations is x tn) = z2 1 e- 2 I- rg2 w-5-7ijk;,-+(t+ iz)2d; (4.7) (4.8) where the error funtion its asymptoti form are given by Erf (u) = -?- f h 0 U eet2 dt 2 1 eeu - l--g u U W (4.9) (4.10) In the limit ~~ -. =J, therefore, we get th e asymptoti : expression for x (n)

16 x o K2 - O3 &qe Z2 i The seond term in the parentheses of Eq. (11) is negligible in the limit (4.11) K2 - =, provided n-k I ---A,,. K2 (4.12) Hene, Eqs. (4) (5) oinide with Eqs. (2.23) (2.33) asymptotially. It is easy to see also that the asymptoti expansion Pn = --L exp - p$] (l+;3$ (yk\ x Gp (4.13) I, +o.,[_ q2-(y2j] Y \ i y )i is valid in the limit y - 03, provided the ondition (4.14) I -I n-m Y2 < ll (4.15)

17 -17- is satisfied. The formulae expressing the parameters of the asymptoti expansions (13) (14) in terms of the mode the moments are idential to those in Setion III. Aknowledgement The author is indebted to Professor S. D. Drell for his warm hospitality extended to him at SC&C. Thanks are also due to W. Colglazier for reading the manusript.

18 -18- Referenes 1. M. G. Kendall A. Stuart, The Advaned Theory of Statistis, Vol. 1 (Charles Griffin Co., London, 1963) J. B. S. Haldane, Biometrika 32, (1942). Y. Tomozawa, Asymptoti Multipliity Distribution Analogue of the Central Limit Theorem, SLAC preprint (1973). 4. G. D. Kaiser, Nul. Phys. B44, (1972); A Comparative Study of Simple Statistial Models for Charged Partile Multipliities, DNPL/P125 (1972); The Approximately Normal Model for Charged Pair Prodution: A Reonsideration, DNPL/P148 (1973) P. Olesen, Phys. Letters 41B, (1972). P. Slattery, Evidene for the Systemati Behavior of Charged Prong Multipliity Distributions in High Energy Proton-Proton Collision, RU-409 (1972). 7. G. W. Parry P. Rotelli, Appliation of the Trunated Gaussian to the Inelasti pp Charged Multipliity Distribution, Trieste, IC/73/3. 8. Y. Tomozawa, Multipliity Distribution in High Energy Collision, SLAC- PUB-1199 (1973). 9. E. A. Cornish R. A. Fisher, Rev. Inst. Int. Statist. 5, (1937); Tehnometris 2, (1960); see also referene 1, Se