THEORY OF STATISTICS.
Tf, as in Chapter XII. §§ 4 et seq. (¢f. especially § 7), a number
of variables are involved, the equations for determining the
coefficients will be given by differentiating
Sabine, pe Tat Dim my 2,2
with respect to each coefficient in turn and equating the result to
zero. This gives the equations of the form there stated. If a
constant term be introduced, its “least square” value will be
found to be zero, as above.
III. THE LAW OF SMALL CHANCES.
(Supplementary to Chapter XV.)
WE have seen that the normal curve is the limit of the binomial
(p +g)" when = is large and neither p nor ¢ very small. The
student’s attention will now be directed to the limit reached
when either p or ¢ becomes very small, but n is so large that
either np or ng remains finite.
Let us regard the n trials of the event, for which the chance of
success at each trial is p, s made up of m +m’ =n trials; then
the probability of having at least m successes in the m +m’
trials is evidently the sum of the m'+1 terms of the expansion
of (p+¢)® beginning with p™ But this probability, which we
may term P,,, can be expressed in another and more convenient
form with the help of the following reasoning. The required
result might happen in any one of m+ 1 ways. For instance :—
(a) Each of the first mm trials might succeed; the chance of
this is p™.
(3) The first m 41 trials might give m successes and 1 failure,
the latter not to happen on the (m + 1)™ trial (a condition already
covered by (a)). But the probability of m successes and 1 failure,
the latter at a specified trial, is p™. ¢, and, as the failure might
occur in any one of m out of m + 1 trials, the complete probability
of (0) is mp™. q.
(¢) The first m + 2 trials might give m successes and 2 failures,
the (m + 2) trial not to be a failure (so as to avoid a repetition
of either of the preceding cases); the probability of this is
m(m+1) mn
aye
In a similar way we find for the contribution of m+ 3 trials,
giving m successes and 3 failures,
m+ 1) +2) ns
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