Application to the Binomial Summation of a Laplacian ^ 

 Method for the Evaluation of Definite Integrals 



By E. C. MOLINA 

 IXTRODUCTIOX 



THE numerical evaluation of the incomplete Binomial Summation, 

 a problem of major importance for many statistical and engi- 

 neering applications of the Theory of Probability, 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 limited range of values of the variables involved ; 

 but a formula of wide applicability is still a desideratum. 



The purpose of this paper is to submit for consideration an approxi- 

 mation formula which seems to meet the situation to a measurable 

 extent. The writer derived it by applying to the equation 



(1) 



^-'^- ^ .v--i(l - xY-^dx 



Jo 



a method which is peculiarly 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 "Theorie Analytique des Probabilites." 

 Poisson applied the method to the integrals in the equation 



r* x"-'/{l -f .r)"+W.v 

 (2) S ( I ) n^ - P) -^ = '^''-''^' 



X—C \ '^ / 



Jq 



/(I -f xY+^dx 



and published a first approximation, together with its derivation, in 

 his "Recherches sur la Probabilite des Jugements." Poisson's ap- 

 proximation seems 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 small numbers." 



1 Presented before International Congress of Mathematicians at Bologna, Italy 

 in September, 1928. 



2 For an excellent resume of some well-known formulas, together with a discussion 

 of their limitations, reference may be had to C. Jordan, "Statistique Mathematique," 

 articles 37 and 38. 



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