# J2 e-*= (27T)- 1 / 2 f V* 1 '»' 1 *»

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1 THE NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION AND A PROOF OF A CONJECTURE OF RAMANUJAN 1 TSENG TUNG CHENG 1. Summary. The Poisson distribution with parameter X is given by (1.1) F(x) = 23 p r where p r = er\ n = [x], for xào. It is well known that F(x) converges to the normal distribution as X» oo. We shall prove the following theorem. THEOREM I. Let *=X- 1/2 (w-x+l/2). Then J2 e-*= (27T)- 1 / 2 f V* 1 '»' 1 *» (1.2) - rl J - + (1/6) (2TTX)- 1 / 2 (1 - x 2 )e-w* 2 + Ô, where 8 satisfies the inequality (1.3) [ Ô <.076X X- 8 / X" 2. This formula is analogous to Uspensky's [l] 2 estimate of the error term in the normal approximation to the binormal distribution and our proof consists in an adaptation of Uspensky's method. At the same time we verify the following conjecture of Ramanujan: for every positive integer n the value of 6 which satisfies the equation ( n n n ~ l n n \ (1.4) \ U - = 1/2 I 1! («1)1 n\) lies between 1/2 and 1/3, and tends to 1/3 as n *<x>. A proof of the above statement was given by Szegö [3]. We give a more elementary proof, using only standard tools, for the following theorem. THEOREM II. For n^7.37 and 1/3. the root of the equation (1.4) in 0 lies between Finally, we shall obtain the following asymptotic expansion of 0: Received by the editors March 15, This paper contains the results of a master's thesis done under the direction of Professor W. Feller and accepted by Cornell University in September, Numbers in brackets refer to the references cited at the end of the paper. 396

2 NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION 397 (1.5) 0 = 1/3 + (4/135)»- 1 + (8/2835)w (rr*). 2. Proof of Theorem I. From the characteristic function we get oo X p r e irt = «*< **-», r«=0 (2.1) f *-«" *&*-»dt = 2x^r. From (2.1) it follows that i> r = (27T)- 1 f'csin^)- 1 r=0 J 0 exp {X(cos * - 1)} sin [(» + 1/2)/ - X sin /]<Ö + (27T)- 1 f (sin //2)- 1 exp {x(cos t - 1)} sin (X sin / + */2)(ft. The second integral is independent of X, since its derivative with respect to X vanishes, as may easily be verified. Putting X = 0 we see the value of the integral is 1/2. Hence we have =(2T)-* f'(siii//2)- rto r\ 2 I Jo (2.2) -exp {X(cos / - 1)} sin [(» + 1/2); - X sin *]<# = /. To estimate the integral I we introduce Then A = (2T)-* f *-*«1 /*(</2)- 1 sin [X»/*a* + X(< - sin 0]<«. / - h\ ^ (2X)- 1 f e-^l\t/2)- l dt (2 3) r T + (27T)- 1 I {r*- 1 ** «/2( sin j/2)-i - er^«1 /«(//2)-i}^ / o = ƒ i + /«+ /a, where the integrals J* are defined and estimated as follows:

3 398 T. T. CHENG [April /i = (2a-)- 1 f {<r 2X 8in2 " 2 (sin t/2)~ l cos */2 - e-^i % {t/2)- 1 }dt (2.4) T = - (27T)- 1 f e-***(t/2)- l dt < 0; (2.5) J, = (2x)- x f e-^i\t/2)- l dt < (32*-)- 1 f Pr^^dt; j t = (27T)- 1 f «-» * " 2 (sin</2)- 1 (l - cos */2)<Ö «/ o < (2*-)- 1 f e~ 2X 8i» 2 " 2 - (sin //2)- 1 (2-6), (sin 2 1/2 cos */2 1 < + (1 - cos tl2) sin 2 t/2\dt < {Sir)- 1 f te-^iht + (32a-)- 1 f *V»«, ''\tt. It follows from (2.3)-(2.6) that I I - /il < (Sir)- 1 f fer** 1 /**» + (327T)- 1 f t*(r**i^dt (2.7) Jo Jo Now we write = (87TX)" 1 + (7T 3 /256)\- 2 < (ZTTX) X~ 2. I x = TT" 1 f r^- x<2 / 2 -sin X^atf-cos X(* - sin t)dt + TT" 1 f r^- x<2 /2-cos X^atf-sin X(/ - sin t)dt (2.8) = K- 1 f rv^«f ' 2 -sinx 1 '»*tó* +/A + JTT- 1 f r^-^'/^cos X 1 ' 2 */- dt + ƒ 5} where = (27T)- 1 / 2 f e~?m + (2TTX)- 1 / 2 (1 - tf 2 )*-* 2 ' 2 + J A + J B, 6 j 4 = Tr-i f r^- x * 2 / 2 [cos X(/ - sin t) - l]<tt,

4 1949] NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION 399 (2.9) / 5 = TT-i f t-h-*' 2 ' 2 cos \ l ' 2 xt- sin X(* - sin /) \dt. Upper limits for / 4 and J 5 are obtained as follows: 2 \ sin t) 2 / 4 <7T" 1 f / 0 2 X ƒ 2 / 6 ri e -^t 2 /2 dt=z (97TX)- 1 ; 0 72 I / 5 i < Tr- 1 f r 1^' 2 ' 2 j[" - X(* - sin *)1 (2.10) + fx(/ sin t) sin \{t sin t) 1 > dt L ^ } V yj j o L J = (2TT)- 1 / 2 X- 3 / 2 [1/ /432] <.043X- 3 ' 2. From (2.7)-(2.10) and the inequality (8X)- 1 + (9TT)- 1 <.076 the required result follows immediately. 3. Proof of Theorem II. Solving (1.4) for 0 and using the formula (2.2), we get (3.1) 26-1 = - r/(n n e~ n /nty, where ƒ' = (27T)- 1 f (sin ] -exp {w(cos t - 1)} < sin f n H ] / w sin / + sin In 1 / n sin / >dt = 7T" 1 I (sin 1 -exp {n(co\$ t 1)} -cos sin n(t sin /)d/ = 27T- 1 f r 1 * «''«-sin n 2 arc sin t(\ J 1*. By Maclaurin expansion, 2 arc sin t/2 - /(l - * 2 /4) 1/2 = / 3 /6 + t*/8q + 3* 7 / t*f(t),

5 400 T. T. CHENG [April where ƒ(t) is an increasing function in the interval 0g/^2. Hence Assume n^7. therefore (3.2) sin n[2 arc sin t/2 - /(l - / 2 /4) 1/2 ] ^ sin (w/ 3 /6 + nt*/80) + 3nf/ nt 9 f(2) <> (nt z /6 + nt 5 /SO) - (nt z /6 + nfi/80)*/6 + (nt*/6 + nfi/soy/120 + Snt 7 / w/ 9. Then the last expression is positive when 2>2, and nt z nt b 1 / nt z nt b \* o I \ 6 80/ + ( 1 H f-.oolint'} dt 120\ 6 80/ 1792 ) < (2 1 rw)- 1 ' 2 {l/ W w ir-«+ 4-1» w~ 6 + 5n-«+ In-- 1 ). In obtaining the last inequality we have, as before, used the familiar relation: 2(2*)- 1 '» f fi*g-vm*dt =1-3 (2* 1). Taking in (3.2) wï; 7 we get (3.3) I' < (2iro) _1/2 (l/3 -.03»- 1 ). On the other hand we can write 2 arc sin t/2 - t(l - P/i) 1 ' 2 = t*/6 + t*g(t) because g(t) increases and g(2) <.057. Hence à t 3 / \ I' > 2X- 1 f r-v-' 1 '""' 2 jsin nÀdt Therefore /.(tit* 1 / nt*\* Ï rv-a/i>»i»j / j _.057nt 6 \dt. (3.4) /' > (2TTW)- 1 / 2 (1/3) -.S05W- 1 ) >.26(2xw)- 1 ' 2, when» è 7. From (3.1), (3.3), (3.4), and Stirling's formula,

6 i 9 49] NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION 401 (2irn) ll2 n n e- n < nl< (2Tfi) 1 f 2 n n e- n+1/12n ) it follows that 20-1< -.26, or 0<.37; and 26-1 > - e 1 / 12w -(27rw) 1 / 2 -(27T^)- 1 / 2 (l/3 -.03/») and therefore 6> 1/3. > - (1 +.09ft- 1 )(1/3 -.03»- 1 ) = - 1/ »- 2 > - 1/3, 4. Proof of the equation (l.s). It can be verified without difficulty that V = (2TT»)- 1 / 2 {1/3-47/540» /6048»" 2 } + 0(»" 7 / 2 ). From a modified form of Stirling's formula due to Feller [2], namely, for w^4, ( Ô).. n\ = (2 )*/*»» exp { -» }, 8 < 1/6, I 12» 360» 3 ; it follows that 20-1 = ~ [1/3-47/540» /6048»" 2 + ] [1 + 1/12» /(2.12»)»-* + ] = - 1/3 + 8/135»" /2835» (»~ 3 ). An obvious reduction then gives the equation (l.s). REFERENCES 1. J. V. Uspensky, Introduction to mathematical probability, New York, W. Feller, On the normal approximation to the binormal distribution, Ann. Math. Statist, vol. 16 (1945) pp G. Szegö, Über einige von S. Ramanujan gestellte Aufgaben, J. London Math. Soc. vol. 3 (1928) pp CORNELL UNIVERSITY

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