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A numerical characteristic of a [[Probability distribution|probability distribution]]. The moment of order  $  k $(
$  k > 0 $
an integer) of a random variable  $  X $
is defined as the mathematical expectation  $  {\mathsf E} X  ^ {k} $, 
if it exists. If  $  F $
is the [[Distribution function|distribution function]] of the random variable  $  X $, 
then

$$ \tag{* }
{\mathsf E} X  ^ {k}  =  \int\limits _ {- \infty } ^  \infty   x  ^ {k}  d F ( x ) .
$$

For the definition of a moment in probability theory, a direct analogy is used with the corresponding idea which plays a major role in mechanics: Formula (*) is defined as the moment of a mass distribution. The first-order moment (a statistical moment in mechanics) of a random variable  $  X $
is the mathematical expectation  $  {\mathsf E} X $. 
The value  $  {\mathsf E} ( X - a )  ^ {k} $
is called the moment of order  $  k $
relative to  $  a $, 
$  {\mathsf E} ( X - {\mathsf E} X )  ^ {k} $
is the central moment of order  $  k $. 
The second-order central moment  $  {\mathsf E} ( X - {\mathsf E} X )  ^ {2} $
is called the [[Dispersion|dispersion]] (or variance)  $  {\mathsf D} X $(
the moment of inertia in mechanics). The value  $  {\mathsf E} | X |  ^ {k} $
is called the [[Absolute moment|absolute moment]] of order  $  k $(
absolute moments are also defined for non-integral  $  k $). 
The moments of the [[Joint distribution|joint distribution]] of random variables  $  X _ {1} \dots X _ {n} $(
see [[Multi-dimensional distribution|Multi-dimensional distribution]]) are defined similarly: For any integers  $  k _ {i} \geq  0 $, 
$  k _ {1} + \dots + k _ {n} = k $, 
the mathematical expectation  $  {\mathsf E} ( X _ {1} ^ {k _ {1} } \dots X _ {n} ^ {k _ {n} } ) $
is called a mixed moment of order  $  k $, 
and  $  {\mathsf E} ( X _ {1} - {\mathsf E} X _ {1} ) ^ {k _ {1} } \dots ( X _ {n} - {\mathsf E} X _ {n} ) ^ {k _ {n} } $
is called a central mixed moment of order  $  k $. 
The mixed moment  $  {\mathsf E} ( X _ {1} - {\mathsf E} X _ {1} ) ( X _ {2} - {\mathsf E} X _ {2} ) $
is called the [[Covariance|covariance]] and is one of the basic characteristics of dependency between random variables (see [[Correlation (in statistics)|Correlation (in statistics)]]). Many properties of moments (in particular, the inequality for moments) are consequences of the fact that for any random variable  $  X $
the function  $  g ( k) =  \mathop{\rm log}  {\mathsf E} | X |  ^ {k} $
is convex with respect to  $  k $
in each finite interval on which the function is defined;  $  ( {\mathsf E} | X |  ^ {k} ) ^ {1 / k } $
is a non-decreasing function of  $  k $. 
The moments  $  {\mathsf E} X  ^ {k} $
and  $  {\mathsf E} ( X - a )  ^ {k} $
exist if and only if  $  {\mathsf E} | X |  ^ {k} < \infty $. 
The existence of  $  {\mathsf E} | X | ^ {k _ {0} } $
implies the existence of all moments of orders  $  k \leq  k _ {0} $. 
If  $  {\mathsf E} | X _ {i} |  ^ {k} < \infty $
for all  $  i = 1 \dots n $, 
then the mixed moments  $  {\mathsf E} X _ {1} ^ {k _ {1} } \dots X _ {n} ^ {k _ {n} } $
exist for all  $  k _ {i} \geq  0 $, 
$  k _ {1} + \dots + k _ {n} \leq  k $. 
In some cases, for the definition of moments, the so-called moment generating function is useful — the function  $  M ( t ) $
with the moments of the distribution as coefficients in its power-series expansion; for integer-valued random variables this function is related to the [[Generating function|generating function]]  $  P ( s ) $
by the relation  $  M ( t) = P ( e  ^ {t} ) $. 
If  $  {\mathsf E} | X |  ^ {k} < \infty $, 
then the [[Characteristic function|characteristic function]]  $  f ( t ) $
of the random variable  $  X $
has continuous derivatives up to order  $  k $
inclusively, and the moment of order  $  k $
is the coefficient of  $  ( it )  ^ {k} / k ! $
in the expansion of  $  f ( t ) $
in powers of  $  t $,

$$ 
{\mathsf E} X  ^ {k}  =  \left . ( - i )  ^ {k} 
\frac{d  ^ {k} }{d t  ^ {k} }
 f ( t) \right | _{t=0} .
$$

If the characteristic function has a derivative of order  $  2 k $
at zero, then  $  {\mathsf E} | X |  ^ {2k} < \infty $.

For the connection of moments with semi-invariants see [[Semi-invariant(2)|Semi-invariant]]. If the moments of a distribution are known, then it is possible to make some assertions about the probabilities of deviation of a random variable from its mathematical expectation in terms of inequalities; the best known are the [[Chebyshev inequality in probability theory|Chebyshev inequality in probability theory]]

$$ 
{\mathsf P} \{ | X - {\mathsf E} X | \geq  \epsilon \}  \leq  \ 

\frac{D X }{\epsilon  ^ {2} }
 ,\  \epsilon > 0 ,
$$

and its generalizations.

Problems of determining a probability distribution from its sequence of moments are called moment problems (cf. [[Moment problem]]). Such problems were first discussed by P.L. Chebyshev (1874) in connection with research on limit theorems. In order that the probability distribution of a random variable  $  X $
be uniquely defined by its moments  $  \alpha _ {k} = {\mathsf E} X  ^ {k} $
it is sufficient, for example, that Carleman's condition be satisfied:

$$ 
\sum_{k=1}^  \infty   
\frac{1}{\alpha _ {2k} ^ {1 / 2 k } }
  =  \infty .
$$

A similar result even holds for moments of random vectors.

The use of moments in the proof of limit theorems is based on the following fact. Let  $  F _ {n} $, 
$  n = 1 , 2 \dots $
be a sequence of distribution functions, all moments  $  \alpha _ {k} ( n ) $
of which are finite, and for each integer  $  k \geq  1 $
let

$$ 
\alpha _ {k} ( n )  \rightarrow  \alpha _ {k} \  \textrm{ as }  n \rightarrow \infty ,
$$

where  $  \alpha _ {k} $
is finite. Then there is a sequence  $  F _ {n _ {i}  } $
that weakly converges to a distribution function  $  F $
having  $  \alpha _ {k} $
as its moments. If the moments determine  $  F $
uniquely, then the sequence  $  F _ {n} $
weakly converges to  $  F $. 
Based on this is the so-called method of moments (cf. [[Moments, method of (in probability theory)|Moments, method of (in probability theory)]]), used, in particular, in mathematical statistics for the study of the deviation of empirical distributions from theoretical ones, and for the statistical estimation of parameters of a distribution (on sampling moments as well as estimators for moments of a certain distribution see [[Empirical distribution|Empirical distribution]]).

====Comments====
The moment generating function is defined by  $  M( t) = {\mathsf E} e  ^ {tX} $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971) Chapt. 1</TD></TR></table>