20° THEORY OF STATISTICS.
binomial distributions. It will have been noted that any one
term—say the 7th—in one series is obtained by taking ¢ times the
rth term together with p times the (r—1)th term of the preceding
series. Now if AP, CR (figure 46) be two verticals, and a third,
BQ), be erected between them, cutting PR in , so that
AB :BC :q:p, then
BQ=p.AP + q.CR.
(This follows at once on joining AR and considering the two
segments into which BQ is divided.) Consider then some
binomial, say for the case p=1, g=2. Draw a series of verticals
(the heavy verticals of fig. 47) at any convenient distance apart
on Bpc
Fre. 46.
on a horizontal base line, and erect other verticals (the lighter
verticals) dividing the distance between them in the ratio of
q:p, viz. 3:1. Next, choosing a vertical scale, draw the binomial
polygon for the simplest case n=1; in the diagram XN has been
taken = 4096, and the polygon is abed, 0b = 3072, 1lc=1024, The
polygons for higher values of » may now be constructed graphically.
 Mark the points where ab, bc, cd respectively cut the
intermediate verticals and project them horizontally to the right
on to the thick verticals. This gives the polygon ad'c’d’e for
n=2. Forob =gq.0b, 1c'=p.0b+q.1c, and so on. Similarly, if the
points where a®’, b'c, etc.,, cut the intermediate verticals are
projected horizontally on to the thick verticals, we have the
polygon ab”¢"d"¢"f” for n=38. The process may be continued

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