300 DOC. 29 DETERMINATION
OF
STATISTICAL VALUES
Doc.
29
Method for the Determination
of
Statistical
Values
of
Observations
Regarding Quantities
Subject
to
Irregular
Fluctuations
by
A.
Einstein
[Archives
des sciences
physiques
et
naturelles
37
(1914):
254-256]
[1]
Suppose
that the
quantity
y
=
F(t) (for example,
the
number of
sunspots)
is
determined
empirically
as a
function of time for
a
very large
interval
T.
How
can one
represent
the statistical behavior of
y?
One
to
this
question, suggested by
the
theory
of
as
follows.
[2]
Suppose
y
is
expanded
as a
Fourier series:
y
=
F(t)
=
5ZA«cos^"^;-
The successive coefficients
An
of the
expansion
will be
very
different from each
other in
magnitude
and
sign,
and will succeed each other in
an irregular manner.
But
2
if
one
forms the
mean
value
A2n
of
An
for
an
interval An of
n
that
is
very large, yet
sufficiently
small for
nAn/T
to
be
very
small,
this
mean
value will be
a
continuous
function of
n.
We will call
it
the
intensity I
of
y corresponding
to
n.
The
intensity
so
defined
will have
a period
= T/n;
we
will
denote it
by
I(0);
the
problem
is to
determine
it.
[3]
A
simple
calculation
yields:
T T
I(0)
=
A^
=
-
J
J
F(t)F(n)'COSirn
-dndt.
T%o
^
From this
it
follows that the function
I
that
we are
seeking might
possibly
be
approximately
determined
up
to
a
numerical factor
by
the
following
rule:
One chooses
a
time interval
A
and
one
forms the
mean
value:
(1)
an
(A)
=
F(t)F(t + A),
which,
for the
given
curve y,
is
a
characteristic function of
A.
For
large
values of
A,
this
curve
will
converge
to
a
limit which could be made
zero
by
a
suitable translation
of the abscissas
(the
axis of
t
and the axis of
A).
Then
one
has:
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