Miscellanea 
139 
On certain Types of Compound Frequency Distributions in which the 
Components can be individually described by Binomial Series. 
By KARL PEARSON, F.R.S. 
Certain difficulties with regard to the interpretation of negative binomials, which are of constant 
occurrence in observational frequency series, have suggested the following investigation. 
Consider a number u of binomial series of which the sth is vg (pa + qs)" and let us suppose 
a frequency series compounded by adding together the rth terms of all these series, such will be 
the compound frequency series it is proposed to discuss. We can realise its nature a little more 
concretely by supposing n balls drawn out of a bag containing Np white and Nq black balls, 
N being very large as compared with n, or else each ball returned before a fresh draw, while 
the values of p and q change discontinuously at the + \, + V2 + "1 + "2 + "3 + 1> stc. 
draws. We shall take as origin the point at which the sum of the first terms of all the binomials 
may be supposed to be plotted. S will denote summation to m terms. 
Let N = S {vg), and Nfi^, Nfi^' be the moments of the compound system aljout this origin, 
Nfji^, N/xo its moments about its mean. Thus 
iV/x/ = S [v^nq,), 
Nfi2 = S {vs (np,q„ + n^q,^)} . 
Hence Nfi^ = nS { v,q, ( 1 + li^ {s {v^q^) - 
= iVju/ + 1^ {nS {v,v,. (q,-qA-\ - NS (v,qf)], 
since N = S (1/,). Accordingly, if a be the standard deviation and tii the moan of the compound 
series, i.e. fx^ = cr^, fj-i = m, then 
M {^."Aq.-qA'} - NS (v,q,') \ 
m N\ S{v,q,) J 
Now suppose we had endeavoured to fit a binomial N {P + Q)" to the compound series, we 
should have had 
m = kQ, (t" = kPQ, 
and accordingly have found 
1 /nS {v,v,, (q, - q,,f} - NS 
N\ S(v,q,) )' ^"^ 
{^("s?.)!'^ 
'\iS{v,v,{q.- q.-f\ - NS{vgq,^) 
Thus, had we attempted to fit a binomial to the heterogeneous series, we should have found 
Q negative and P greater than unity provided 
nS{v,vAq.- q.'f\ he > NS (v,q,^), 
a condition which will frequently be found to be satisfied, especially if q^ be small and 71 large. 
In the limit let us take nq^ = nig and g., vanishingly small, i.e. suppose the sth binomial to be 
replaced by the Poisson series 
then we have at once 
NS {v^m,} 
N8(v,m,) ■ } (iv) 
(S {v^m,]) 
Thus, if two or more Poisson's series be combined term by term from, fh" first, then the 
compound will always be a negative binomial. This theorem was first pointed out to me by 
"Student" and suggested by him as a possible explanation of negative binomials occurring in 
