xxxiv Tables for Statisticians and Biometricians [III 



We have then : 



^ = -5 + T V -380,2012 (1 - -00195,5437 + -00000,3576) 

 = 5 + -031,6216 = -531,6216. 



Again *i = -2371,3534, 



(9Q71 QKQA 1 



and therefore f+1 = u t j T O ^ 26 55 + ' 531 > 6216 f 



= u t x 3-020,8264. 



450! /l\ 190 /2\ 260 

 Now ^ = ^=T90T260!l3J \f) 



= 1-710072 x 10~ 5 . 

 Accordingly S t+1 = 1-710072 x 3'020,8264 x 10~ 5 



= 5-1658 x 10- 5 . 



The true value found by adding the terms, retaining nine decimals, is 

 51662 xlO- 5 f. 



Illustration (iv). Not infrequently this method of summing + 1 terms of a 

 binomial will give good results when we have to evaluate an Incomplete B-function 

 lying outside the range of our tables, where the dichotomic ordinate lies between 

 1'5 and 3'0 times the standard deviation from the mode. 



If B x (p, q) = f V- 1 (1 - xy-idx, 



Jo 



ri 

 and B (p, q) = a?' 1 (1 - as^das, 



Jo 



then B x (p, q)(B (p, q) is the Incomplete B-function Ratio and is the probability 

 integral of the Type I curve y = y x p ~ 1 (l x)i~\ 



Now we know J that 



x P, q) _ i _ t ne sum of the first p terms of the binomial ((1 - x} + x) n (vii), 



where n=p + q 1. 



But ((1 - a?) + x) n = 1. Hence 



,-f - = ((1 a?) + x} n the sum of the first p terms 

 B (p, q) 



= the sum of the last q terms, 



or n/ = t ne sum of the first q terms of the binomial ( + (! #))"...( viii). 



B (P> V 



* ^(l-a a;i ) = -0000,3290 > ?45 ) *., = -0001,3874, 967. 



t Had we used, as is not infrequently done, a normal curve to fit the whole binomial (for it has 

 the high power of 450), the answer would have been 3'91 x 10~ 6 ! 



* Biometrika, Vol. xvi. p. 202. 



