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
answer
to
this
question, suggested by
the
theory
of
radiation, is
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:
Previous Page Next Page

Extracted Text (may have errors)


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
answer
to
this
question, suggested by
the
theory
of
radiation, is
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:

Help

loading