<s>
In	O
probability	O
theory	O
and	O
statistics	O
,	O
a	O
central	B-General_Concept
moment	I-General_Concept
is	O
a	O
moment	O
of	O
a	O
probability	O
distribution	O
of	O
a	O
random	O
variable	O
about	O
the	O
random	O
variable	O
's	O
mean	O
;	O
that	O
is	O
,	O
it	O
is	O
the	O
expected	O
value	O
of	O
a	O
specified	O
integer	O
power	O
of	O
the	O
deviation	O
of	O
the	O
random	O
variable	O
from	O
the	O
mean	O
.	O
</s>
<s>
Central	B-General_Concept
moments	I-General_Concept
are	O
used	O
in	O
preference	O
to	O
ordinary	O
moments	O
,	O
computed	O
in	O
terms	O
of	O
deviations	O
from	O
the	O
mean	O
instead	O
of	O
from	O
zero	O
,	O
because	O
the	O
higher-order	O
central	B-General_Concept
moments	I-General_Concept
relate	O
only	O
to	O
the	O
spread	O
and	O
shape	O
of	O
the	O
distribution	O
,	O
rather	O
than	O
also	O
to	O
its	O
location	O
.	O
</s>
<s>
Sets	O
of	O
central	B-General_Concept
moments	I-General_Concept
can	O
be	O
defined	O
for	O
both	O
univariate	B-General_Concept
and	O
multivariate	O
distributions	O
.	O
</s>
<s>
The	O
nth	O
moment	B-General_Concept
about	I-General_Concept
the	I-General_Concept
mean	I-General_Concept
(	O
or	O
nth	O
central	B-General_Concept
moment	I-General_Concept
)	O
of	O
a	O
real-valued	O
random	O
variable	O
X	O
is	O
the	O
quantity	O
μn	O
:=	O
E[( XE[X]	O
)	O
n ]	O
,	O
where	O
E	O
is	O
the	O
expectation	O
operator	O
.	O
</s>
<s>
For	O
random	O
variables	O
that	O
have	O
no	O
mean	O
,	O
such	O
as	O
the	O
Cauchy	O
distribution	O
,	O
central	B-General_Concept
moments	I-General_Concept
are	O
not	O
defined	O
.	O
</s>
<s>
The	O
first	O
few	O
central	B-General_Concept
moments	I-General_Concept
have	O
intuitive	O
interpretations	O
:	O
</s>
<s>
The	O
"	O
zeroth	O
"	O
central	B-General_Concept
moment	I-General_Concept
μ0	O
is	O
1	O
.	O
</s>
<s>
The	O
first	O
central	B-General_Concept
moment	I-General_Concept
μ1	O
is	O
0	O
(	O
not	O
to	O
be	O
confused	O
with	O
the	O
first	O
raw	O
moment	O
or	O
the	O
expected	O
value	O
μ	O
)	O
.	O
</s>
<s>
The	O
second	O
central	B-General_Concept
moment	I-General_Concept
μ2	O
is	O
called	O
the	O
variance	O
,	O
and	O
is	O
usually	O
denoted	O
σ2	O
,	O
where	O
σ	O
represents	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
The	O
third	O
and	O
fourth	O
central	B-General_Concept
moments	I-General_Concept
are	O
used	O
to	O
define	O
the	O
standardized	O
moments	O
which	O
are	O
used	O
to	O
define	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
,	O
respectively	O
.	O
</s>
<s>
The	O
nth	O
central	B-General_Concept
moment	I-General_Concept
is	O
translation-invariant	O
,	O
i.e.	O
</s>
<s>
For	O
all	O
n	O
,	O
the	O
nth	O
central	B-General_Concept
moment	I-General_Concept
is	O
homogeneous	O
of	O
degree	O
n	O
:	O
</s>
<s>
A	O
related	O
functional	O
that	O
shares	O
the	O
translation-invariance	O
and	O
homogeneity	O
properties	O
with	O
the	O
nth	O
central	B-General_Concept
moment	I-General_Concept
,	O
but	O
continues	O
to	O
have	O
this	O
additivity	O
property	O
even	O
when	O
n≥4	O
is	O
the	O
nth	O
cumulant	O
κn(X )	O
.	O
</s>
<s>
For	O
n	O
=	O
1	O
,	O
the	O
nth	O
cumulant	O
is	O
just	O
the	O
expected	O
value	O
;	O
for	O
n	O
=	O
either	O
2	O
or	O
3	O
,	O
the	O
nth	O
cumulant	O
is	O
just	O
the	O
nth	O
central	B-General_Concept
moment	I-General_Concept
;	O
for	O
n≥4	O
,	O
the	O
nth	O
cumulant	O
is	O
an	O
nth-degree	O
monic	O
polynomial	O
in	O
the	O
first	O
n	O
moments	O
(	O
about	O
zero	O
)	O
,	O
and	O
is	O
also	O
a	O
(	O
simpler	O
)	O
nth-degree	O
polynomial	O
in	O
the	O
first	O
n	O
central	B-General_Concept
moments	I-General_Concept
.	O
</s>
<s>
For	O
the	O
cases	O
n	O
=	O
2	O
,	O
3	O
,	O
4	O
—	O
which	O
are	O
of	O
most	O
interest	O
because	O
of	O
the	O
relations	O
to	O
variance	O
,	O
skewness	B-General_Concept
,	O
and	O
kurtosis	B-Error_Name
,	O
respectively	O
—	O
this	O
formula	O
becomes	O
(	O
noting	O
that	O
and	O
)	O
:	O
</s>
<s>
In	O
distributions	O
that	O
are	O
symmetric	O
about	O
their	O
means	O
(	O
unaffected	O
by	O
being	O
reflected	B-Algorithm
about	O
the	O
mean	O
)	O
,	O
all	O
odd	O
central	B-General_Concept
moments	I-General_Concept
equal	O
zero	O
whenever	O
they	O
exist	O
,	O
because	O
in	O
the	O
formula	O
for	O
the	O
nth	O
moment	O
,	O
each	O
term	O
involving	O
a	O
value	O
of	O
X	O
less	O
than	O
the	O
mean	O
by	O
a	O
certain	O
amount	O
exactly	O
cancels	O
out	O
the	O
term	O
involving	O
a	O
value	O
of	O
X	O
greater	O
than	O
the	O
mean	O
by	O
the	O
same	O
amount	O
.	O
</s>
<s>
The	O
2nd-order	O
central	B-General_Concept
moment	I-General_Concept
β2	O
is	O
called	O
the	O
variance	O
of	O
X	O
whereas	O
the	O
2nd-order	O
central	B-General_Concept
moment	I-General_Concept
α2	O
is	O
the	O
pseudo-variance	O
of	O
X	O
.	O
</s>
