DOC.
7
PROBABILITY CALCULUS
215
Thus,
the number $
is
composed
of
two
parts: a $1,
which
comes
from
the
summand
-S/2Z,
and
a $2,
which
comes
from
f(a)/y[Z.
b1
contains
all
those
S
that
have
been
at
a
positive
distance
S0/2Z
from
the
value
,S0;
and,
to
be
sure,
these members
cross
S0
in
the
negative
direction. Since
S/2Z is
a
very
small
number,
the number of these members
is,
up
to
infinitely
small
quantities
of
higher
order,
(8)
% =
-|pPo)-
Contributions
to
the number
b2
come
from
every
arbitrary
positive
and
negative
distance
A
from
S0;
indeed,
the contributions
are
positive
or
negative,
depending on
whether
A
is
negative or positive.
The number dN
at
the
distance
A
is
given by
F(S0
+
A
)dS
=
F(S0 +
A)dA,
or,
since
only
small values
of
A
are,
after
all,
of
importance,
by
+
A(~)
dA
0
Of
this
number,
all
those
cross
the
value
S0
in
the
positive
direction
which, coming
from
a
negative A,
have
a
f(a)
so large
that
f(a)
a
141,
and thus the number
will
be
J
CP(f)df.
Analogously,
the number
going
in
the
negative
direction
will be
-Vz
J
P(f)df.
We will
then
have