In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions.

As its name implies, the moment generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0.

In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases.

The moment-generating function of a real-valued distribution does not always exist, unlike the characteristic function. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments.

Contents

DefinitionEdit

The moment-generating function of a random variable X is

 

wherever this expectation exists. In other words, the moment-generating function is the expectation of the random variable  . More generally, when  , an  -dimensional random vector, and   is a fixed vector, one uses   instead of  :

 

  always exists and is equal to 1. However, a key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead.

The moment-generating function is so named because it can be used to find the moments of the distribution.[1] The series expansion of   is

 

Hence

 

where   is the  th moment. Differentiating     times with respect to   and setting  , we obtain the  th moment about the origin,  ; see Calculations of moments below.

If   is a continuous random variable, the following relation between its moment-generating function   and the two-sided Laplace transform of its probability density function   holds:

 

since the PDF's two-sided Laplace transform is given as

 

and the moment-generating function's definition expands (by the law of the unconscious statistician) to

 

This is consistent with the characteristic function of   being a Wick rotation of   when the moment generating function exists, as the characteristic function of a continuous random variable   is the Fourier transform of its probability density function  , and in general when a function   is of exponential order, the Fourier transform of   is a Wick rotation of its two-sided Laplace transform in the region of convergence. See the relation of the Fourier and Laplace transforms for further information.

ExamplesEdit

Here are some examples of the moment-generating function and the characteristic function for comparison. It can be seen that the characteristic function is a Wick rotation of the moment-generating function   when the latter exists.

Distribution Moment-generating function   Characteristic function  
Degenerate      
Bernoulli      
Geometric    
 
 
Binomial      
Negative Binomial      
Poisson      
Uniform (continuous)      
Uniform (discrete)      
Laplace      
Normal      
Chi-squared      
Noncentral chi-squared      
Gamma      
Exponential      
Multivariate normal      
Cauchy   Does not exist  
Multivariate Cauchy

 [2]

Does not exist  

CalculationEdit

The moment-generating function is the expectation of a function of the random variable, it can be written as:

Note that for the case where   has a continuous probability density function  ,   is the two-sided Laplace transform of  .

 

where   is the  th moment.

Linear combination of independent random variablesEdit

If  , where the Xi are independent random variables and the ai are constants, then the probability density function for Sn is the convolution of the probability density functions of each of the Xi, and the moment-generating function for Sn is given by

 

Vector-valued random variablesEdit

For vector-valued random variables   with real components, the moment-generating function is given by

 

where   is a vector and   is the dot product.

Important propertiesEdit

Moment generating functions are positive and log-convex, with M(0) = 1.

An important property of the moment-generating function is that if two distributions have the same moment-generating function, then they are identical at almost all points.[3] That is, if for all values of t,

 

then

 

for all values of x (or equivalently X and Y have the same distribution). This statement is not equivalent to the statement "if two distributions have the same moments, then they are identical at all points." This is because in some cases, the moments exist and yet the moment-generating function does not, because the limit

 

may not exist. The lognormal distribution is an example of when this occurs.

Calculations of momentsEdit

The moment-generating function is so called because if it exists on an open interval around t = 0, then it is the exponential generating function of the moments of the probability distribution:

 

That is, with n being a nonnegative integer, the nth moment about 0 is the nth derivative of the moment generating function, evaluated at t = 0.

Other propertiesEdit

Jensen's inequality provides a simple lower bound on the moment-generating function:

 

where   is the mean of X.

Hoeffding's lemma provides a bound on the moment-generating function in the case of a zero-mean, bounded random variable.

When all moments are non-negative, the moment generating function gives a simple, useful bound on the moments:

 

The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X. This statement is also called the Chernoff bound. Since   is monotonically increasing for  , we have

 

for any   and any a, provided   exists. For example, when X is a standard normal distribution and  , we can choose   and recall that  . This gives  , which is within a factor of 1+a of the exact value.

Relation to other functionsEdit

Related to the moment-generating function are a number of other transforms that are common in probability theory:

Characteristic function
The characteristic function   is related to the moment-generating function via   the characteristic function is the moment-generating function of iX or the moment generating function of X evaluated on the imaginary axis. This function can also be viewed as the Fourier transform of the probability density function, which can therefore be deduced from it by inverse Fourier transform.
Cumulant-generating function
The cumulant-generating function is defined as the logarithm of the moment-generating function; some instead define the cumulant-generating function as the logarithm of the characteristic function, while others call this latter the second cumulant-generating function.
Probability-generating function
The probability-generating function is defined as   This immediately implies that  

See alsoEdit

ReferencesEdit

CitationsEdit

  1. ^ Bulmer, M. G., Principles of Statistics, Dover, 1979, pp. 75–79.
  2. ^ Kotz et al. p. 37 using 1 as the number of degree of freedom to recover the Cauchy distribution
  3. ^ Grimmett, Geoffrey (1986). Probability - An Introduction. Oxford University Press. p. 101. ISBN 978-0-19-853264-4.

SourcesEdit