Karl Pearson 
11 
This is the curve whicli corresponds as closely to the skew binomial histogram 
as the normal curve has been shown by Laplace to correspond to the symmetrical 
binomial histogram, or to the skew binomial even if n be very large and the bases 
of the histogram very small. 
In general however when none of these conditions hold we have 
1 dy _ X 
where a„ = PQ (iii + L) ( 1 + — ) C" and is 'positive, 
«i = {Q — P) ^ + c and may be of either sign, 
cu = — ^ and is negative. 
Thus for this pai'ticular case we may throw the result into the foi'm 
\dy__ _ J ( K h ] 
y dx ~ " h (lh~x){b., Vx) ~ h,(b, + b.^ \b, + x b, - x\ ' 
Hence 
log, ?/ = constant + ^ (^/^^/ ^ f^'-i + ■''') + ^'i 1"8> ('^i " 
J . 6i bo 
Let s, = , — , , . . So = 
Thus ^ = y„(l+£y^(l_ 
where i/„ is the modal ordinate. 
This is a limited range curve, of which the partial area is expressible in terms 
of the incomplete i?-function. In other words, the so-called " probability integral " 
is only a very special case of the incomplete i?-function, which is in this sense the 
general probability integral. 
We easily find 
b = b, + b, = c l^^i) + m + l- n ^PQ]^ n + in + l + v ^/'PQ\K 
and b,-b.,^c{Q- P){lu + m +l). 
Thus b, = -^b + lc{Q- P) (i n + m + 1), 
b., = \b- U {Q - P) {hi + m + 1). 
Hence s, = nb^jb 
1,, (i + {Q-P){ln + vi-^l) 
[i ji + m + l-n VPQP [^n + m + l + n \/PQy 
(Q-P) (in + m + l) 
Similarly = 2" ( 1 ~ 
[in + m + 1 - n \/PQf {hi + + 1 + « ^PQV 
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