DOC.
7
PROBABILITY CALCULUS
213
t t
fi(oL),f2(a)
... (corresponding
to
sin
2zn,cos
2cni).
We
must
subject
these
n
functions to
an
additional restriction:
namely,
from
the
probability
that
one
of
the
quantities
a
lies
between
a +
da there
follows
a
statistical law
for the
f;
let the
probability
q(f)df
that
f
has
a
numerical
value
between
f
and
f +
df
be
always
such
a
function
that the
average
value
f
 =
0.
(It
can easily
be
seen
that
our
functions sin and
cos
indeed
fulfill this
assumption,
because
if
every
value
of
ts
between
0
and
T
is
equally
probable,
the
average
values
t
sin
2tn

T
and
cos
2tn

T
vanish.)
We
now
assemble
a
(very
large)
number Z of
such
elements
a
into
one
system.
To
such
a system
belong
certain
sums
E(z)/i(a)
/,(«)•••
(corresponding
to
the
coefficients
A/an,
Bn/an).
We
set ourselves
the task of
finding
the
statistical law
that
a
combination of these
sums obeys.
First
we
must
be clear about
a
fundamental
point.
The
statistical law
obeyed
by
the
sums
23
themselves
will
not
at
all
be
independent
of the number Z of the elements.
This
we can
easily
see
in
the
simple special
case
when
f(a)
can assume only
the
values
+1
and
1. Then
we evidently
have
and
y =y
±i
£i{Z+1)
^(Z)
7
=
y
+1.
i/(Z1)
Thus,
the
mean square
value
of the
sum
increases
proportionally
to
the number of
ele
ments.
Hence,
if
we
wish to arrive at
a
statistical law
that
is independent
of
Z,
we
must
not
consider the
E
but
rather,
since
T,2/Z
stays
constant,
the
quantities
si.