E. C. MOLINA (New York - U. S. A) APPLICATION TO THE BINOMIAL SUMMATION OF A LAPLACIAN METHOD FOR THE EVALUATION OF DEFINITE INTEGRALS Introduction. The numerical evaluation of the incomplete Binomial Summation, a problem of major importance for many statistical and engineering applications of the Theory of ProbabiKty, is a question for which a satisfactory solution has not as yet been obtained. Several approximation formulas have been presented (*), each of which gives good results for some Kmited range of values of the variables involved; but a formula of wide applicabikty is still a desideratum. The purpose of this paper is to submit for consideration an approximation formula which seems to meet the situation to a measurable extent. The writer derived it by applying to the equation p _ / a;, '~ (l x)' l ~''dx (i) s t) px{ - p)n ~~ x=? -i ' X==G /V - (l x)"- ( 'dx d a method which is pecukarly efficacious for approximately evaluating definite integrals when the integrands contain factors raised to high powers. The method used constitutes the subject matter of Chapter I, Part II, Book I of Laplace's, Théorie Analytique des Probabilités. POISSON applied the method to the integrals in the equation fx'^'/il + xy'+^dx () s ÇW -^œ= ^ x ~ c fx n^/(l + x) n + i dx ó and published a first approximation, together with its derivation, in his Recherches sur la Probabilité des Jugements. Poisson's approximation seems (*) For an excellent resumé of some well-known formulas, together with a discussion of their limitations, reference may be had to C. JORDAN : Statistique Mathématique, articles 7 and.
COMUNICAZIONI never to have been used and was less fortunate than his famous limit to the binomial expansion which also was lost sight of until it reappeared under the caption «law of smak numbers v. While the integrals in equations () and () are wek known equivalent forms for the complete and incomplete Beta functions, the equations themselves are not so famikar although one or the other wik be found in LAPLACE, POISSON, BOOLE (Differential Equations) and at least two other places. Approximate Formula. The approximate formula derived from equation () and submitted herewith for consideration la T () Sß^-^-sKlK' where Si is the i th approximation to the infinite series B,T*-*[ + (s- DV-* + (a-)(»- )7\-...] () i=i, + S [l 5... (s l)]- s s l T, = Ti, (5) r» = ( B _i)iog_^_ +(c_i)iog c J + ( w _ e )iog?-j», and a=np\ T to be taken negative when a<(c l)n/(n l). The first, second and third approximations to the infinite series S are where Q T? Q i + B T Q B L -\-B T-\-B {l-j- T ) i =[(w-c)-(c-l)]/f / (w-l)(to-c)(c-l), R = (l/)[l/(n-e) + l/(e-l)-/(n-l)], R =- (/5) J R [ J R + /(n -)]. It will be noted that R», \R ± \ and i are symmetric functions of (n o) and (c ). For the limiting case (Poisson's Exponential Binomial Limit) where ra=oo, jp= but np = a, we have T* = l + (c-l)log(c-l)/a+(a-c). Ä t =/^=), JB,=/(C-), ^=-(/5)^^.
E. C. MOLINA : Evaluation of definite integrals 7 Numerical Results. Since it is easy to compute the binomial summation directly when either c or n c is small, the practical value of an approximate formula depends on its efficiency for large values of these quantities. The analysis given below under the beading «Derivation of the Approximate Formula» indicates that the successive R s 's in the series for S decrease when ic and in c increase. Therefore, when these two quantities are large, a few terms of the approximate formula () may be expected to give satisfactory results. As a matter of fact, the formula gives good results when ^c and \n c are not large. To confirm this statement the Tables ( L ) given at the end of this paper are submitted. In the th column of each table are given times the true values of x=n In P^QPHI-PY -*- In the columns headed A i9 A and A are given times the differences between the true values and those obtained by applying formula () with the first, second and third approximations to S respectively. Table I in Czuber's, Wahrscheinlichkeitsrechnung, was used for evaluating the probabkity integral in equation (). The range of values of P covered by the tables is such that at the lower end of each section P>.5 while at the upper end P<.9995, except where this latter condition would call for a value of c<. Of course, a larger or smaker range might have been given. The decision as to this question was based on the fact that several writers on the theory of statistics, when dealing with the normal law of errors, speak of an error exceeding or times the standard deviation as being a very improbable event. In order to keep the number of pages required for the tables within reasonable bounds computations were made only for even values of c. The values of a=np used are such that each of the values p = l/, p = l/ and p = l/ occurs twice; likewise each of the values n=, n=5 and ^= occurs twice. A greater degree of accuracy than that indicated by the tables can, of course, be obtained by working out and using i, JR 5,...; for this purpose, recourse should be had to equation () below and the details immediately fokowing it. The only practical limitation to the use of formula () would appear to be the number of places given by the existing tables for the probability integral. (*) I am greatly indebted to Miss NELLIEMAE Z. PEARSON of the Department of Development and Research both for supervising the work of my computers and contributing personally several sections of the tables.
COMUNICAZIONI However, this difficulty is encountered only when P, or ( P), is smak, in which case T is large and the integral T [er t: dt may be readily evaluated by computing the first few terms of the series [e- T '/T^][l-Tr + (l )Tr -(l. ö)^-*...], where, as above, Ti = Ttf. When Pis very small, the difference c a=c np is relatively large compared to a, and for this latter case recourse may be had to the approximate formula published by the writer in the American Mathematical Monthly for June, 9. Derivation of the Approximate Formula. FoKowing LAPLACE closely, let us set () y(x) = Ye-t% where Y=y(x Q ) is the maximum value of y(x). Then (7) fydx=y)e-^)dt, oo the upper limit T being given by the equation () y&)=y(z*)e- T \ Assuming dx/dt expanded in powers of t so that (9) dx/dt==^d, q+i t s and setting R s =D s+i /D i, equation (7) reduces to p [ydx= YD, R s fpér*'dt. Ó s== Our fundamental equation () may now be written T x=~n ^B s j t s e~ r 'dt < > ïtw-^- -^ - s= oo
E. C. MOLINA: Evaluation of definite integrals 9 Integrating by parts and separating the terms involving (e-t'dt from the terms containing e~ v, we obtain equations () and (). To determine R s =D s+i /D l, note that equation () gives t=(log Y logy) i and set v(x) = (x x )/(log Y logy)! so that x may be written in the form x=x + v(x)t. This form for x gives the expansion (Lagrange's Theorem for the simple case where f(x)=x; see Modern Analysis by WHITTAKER and WATSON) _ V ** ( d *~ iv '\ x ~^osi\^=ï) x^q- Comparing this expansion for x with the previous expansion (9) for dx/dt, we obtain D i =v(x ) aild A + / l d^h\ s \\v(x) daf ) x = Xo ' D i Up to this point no particular form has been attributed to the function y(x). From now on we deal with the function which constitutes the integrand of the integrals in equation (). The function.. t/a y(x)=x c - i (l-x) n - c gives the expansion (log Y logy) = (x Xo) [Ao + A l (x x ) + A (x x Q )...], where. x = (c-l)/(n-l) is the value of x for which y(x) is a maximum and. l \d s + {\o g Y-logij) 9 (*+)! dx'+ Jx=x Q We are now prepared to evaluate R s. g=a + Ai(x x ) + A (x x Q )... and 7o, *» g s =d*g/dx s. Then Set g=(/)^/ [(5/)^-^],
COMUNICAZIONI Therefore, since g s =sla s when x=x, P = (/)^-[(5/)^f-^^], R = -A^I [A A -A A i A + Afl Substituting for A, Ai, A and A the expressions derived by giving s the values,, and respectively in equation (), we obtain for R i9 R and R the functions of n and c given on page. For values of s greater than the direct evaluation of d v s+i ldx? by successive differentiation becomes very tedious. It will be found much more practical to use the fokowing procedure (*), where D is a symbol of operation, A=A, and b=a ± /d*a~( s+i) ' \ da-is+i)li UdA D s ~ b + or () + \da* D* ô +... + ds-, A -(S+i)ß (s l)\da'-\ d>n A -(s+d/ (Ds-mfyn). m! da» w=l Db '^ rfm-(s+d/ slda* The fokowing equations give the details requisite for the formation of R s to R inclusive; A s can be computed from equation (). Db=A, D b=a, D b=a, D*b=A 5, D 5 b=a, D«b=A, D b=a s, Db =A i A, D b =A i A + A\, D*b =A i A i + A A, D i b =A i A, + A A é + A, I) 5 b =A i AB + A A 5 + A A D*b =A i Ai + A A + A As + A, Db* = A A, D b = SA A + SA L A, D b =SA A i + A i A A + AI, D^ = SA A 5 + A i A A + SA i Al + SA A, D 5 b = SA A + A L A A b + A i AA i + SA A i + SA A, Db* = ±A\A, ( l ) See De Morgan's: Differential and Integral Calculus, () page, art..
E. C. MOLINA: Evaluation of definite integrals ZW=AlA + AtAî, D i b i =AAfA 5 + AlA A i + QAlAl+A l AtA,. + At, Db*=òAtA, D*b*=5A\A + \.A\A\, DW=5AiA i + AlA î A + AlAl, Db s =A{A, D b 5 =AlA s +5 A\A\, DW=A\A,_. or To illustrate the use of the procedure given above, let us evaluate R. We have *-*- m) *>+$ )»»+{% ) +^> = - (5/) Afl*(A t ) + (I)(5/)(J/)A^I*(A A, + AÏ) -(l/)(5/)(7/)(9/)^-"/wm t ) + (l/)(5/)(7/)(9/)(ll/)il -»/»^î R i -*(5/)A;*[-AlA t + (7/)Al(A i A, + Al/) - (l/)(7/)(9/) A AtA + (/) (7/) (9/) (/) At]. Tables Indicating Degree of Accuracy Obtainable by Use of Formula () for Evaluating. X=C Pi = lst approximation, A l = (P P l ) 5, P =d approximation, A = (P P ), P =d approximation, A = (P P ) G, a=np, Z"=(n-) log^ +(d-l) log ^ +(n-c) log ^, 7=-^ rut! r ferodi.
COMUNICAZIONI TABLE I. /(IO ) P( c ) ^ a =.5; n = oc; p-= A +.75.7.575.95 5 9 577 7 5 5 7 599 95 99 a =.5; ^=; # =.5 +.5.9.99.7 57 5 5 77 5 - - 7 57 9 5 97 7 TABLE II. 7( tì ) P( ) A a = = 5 ; n = c.5.75..5975.59.9.777.55 95 5 99 55 5 7 59 95957 797 7 55 9 9 9 7 59 97 79 _ 7 5 a 5; n=; p=m.5597.9.7..9.757.799.75 995 97 97 9 77 9 79 99 7 79 7 7 7 9 ; 9 i 99, 5 75 7 5 7* a = = 5; w = 5; p =.l.5775..57.9.57.7999.9.7595 975 7 7 75 5 57 5 9 797 77 5 5 5 7 9 599 79 7 5 9 7 7 95 7 5 \p-p \>\p-p,\
E. C. MOLINA: Evaluation of definite integrals TABLE III. T ( G ) P( ) A*, «= ; n = DO; p =.579.7.5.799.79.7...5.7759.5. 9997 995 99 5 7 77 75 7 99 577 5 99999 99 99 77977 59 555 7 77 5 99 9 7 5 5 55 79 7 77 7 5 7 75 5 57 5 9 5 7 7 5 5 7 - - a- = ; w = ; p = A.597.979.755.7.9.5.7.595.555.9.757.7 9999 99 99 99 5 59 59 5 5 5 99979 99 9 7999 57 997 77 99 7 979 5 7 997 5 7 7 7 9 79 9 77 7 5 - * 9* 5 9 9 \P-P \ > \P-P i \ TABLE IV e T J( ) P( ) ^ ^ a = 5; w = oo; jt? =.7.955...77.775..575.7999.5.59.5.9795.7 9999 99 995 959 975 75 5 59 75 7 7 97 7 9 9997 9977 999 97 59 7 9 5 7 5 9 75 is 55 9 55 5 5 9 5 5 5 7 5 9 9 g 5
COMUNICAZIONI TABLE IV. (Continued) a = 5; w=; p=.5.955..57.979.97.95.57.9575.777.577.9977 999 9975 97 977 7 5 7597 5979 99 555 9997 997 97 9975 777 77 797 99 75 5 559 7 59 55 9 9 5 9 5 7 7 97 7 9 7 _ 77 9 5 7 9 TABLE V. ( ) P( ) = 5; = oo: p = :.97.759.59.55.97.575.755..7999.5555.759.779.7.595..9.55 999 9999 9959 97 9555 999 7977 575 5 77 79 7 97 59 7 55 99977 995 995 97775 995 7597 77 99 5 7 97 9 5 7 7-5 59 7 75 5 5 7 7 7 5 5 9 9 5 59 9 9 9 75 5 9 7 9 7 7 7 a = 5; w==5; p =.h.97.977.5779.59.7. -. -.555 -.97..7577.555.597 99959 997 9 975 759 59 7 9 59 7 7 9995 9999 95 959 9 99 9 5 77 5 9 7 5 7 99 5 77 9 57 7 57 9 9 7 9 5 9 7 7 95 5 7