Moment-generating function

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.

Definition

The moment-generating function of a random variable X is

M_{X}(t):=\operatorname {E} \left[e^{tX}\right],\quad t\in \mathbb {R} ,

wherever this expectation exists. In other words, the moment-generating function is the expectation of the random variable $e^{tX}$ . More generally, when $\mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }$ , an $n$ -dimensional random vector, and $\mathbf {t}$ is a fixed vector, one uses $\mathbf {t} \cdot \mathbf {X} =\mathbf {t} ^{\mathrm {T} }\mathbf {X}$ instead of $tX$ :

M_{\mathbf {X} }(\mathbf {t} ):=\operatorname {E} \left(e^{\mathbf {t} ^{\mathrm {T} }\mathbf {X} }\right).

$M_{X}(0)$ 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 $e^{tX}$ is

e^{t\,X}=1+t\,X+{\frac {t^{2}\,X^{2}}{2!}}+{\frac {t^{3}\,X^{3}}{3!}}+\cdots +{\frac {t^{n}\,X^{n}}{n!}}+\cdots .

Hence

{\begin{aligned}M_{X}(t)=\operatorname {E} (e^{t\,X})&=1+t\operatorname {E} (X)+{\frac {t^{2}\operatorname {E} (X^{2})}{2!}}+{\frac {t^{3}\operatorname {E} (X^{3})}{3!}}+\cdots +{\frac {t^{n}\operatorname {E} (X^{n})}{n!}}+\cdots \\&=1+tm_{1}+{\frac {t^{2}m_{2}}{2!}}+{\frac {t^{3}m_{3}}{3!}}+\cdots +{\frac {t^{n}m_{n}}{n!}}+\cdots ,\end{aligned}}

where $m_{n}$ is the $n$ th moment. Differentiating $M_{X}(t)$ $i$ times with respect to $t$ and setting $t=0$ , we obtain the $i$ th moment about the origin, $m_{i}$ ; see Calculations of moments below.

If $X$ is a continuous random variable, the following relation between its moment-generating function $M_{X}(t)$ and the two-sided Laplace transform of its probability density function $f_{X}(x)$ holds:

M_{X}(t)={\mathcal {B}}\{f\}(-t),

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

{\mathcal {B}}\{f_{X}\}(s)=\int _{-\infty }^{\infty }e^{-sx}f_{X}(x)\,dx,

and the moment-generating function's definition expands to

M_{X}(t)=\operatorname {E} \left[e^{tX}\right]=\int _{-\infty }^{\infty }e^{tx}f_{X}(x)\,dx.

This is consistent with the characteristic function of $X$ being a Wick rotation of $M_{X}(t)$ when the moment generating function exists, as the characteristic function of a continuous random variable $X$ is the Fourier transform of its probability density function $f_{X}(x)$ , and in general when a function $f(x)$ is of exponential order, the Fourier transform of $f$ 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.

Examples

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 $M_{X}(t)$ when the latter exists.

Distribution	Moment-generating function $M_{X}(t)$	Characteristic function $\varphi (t)$
Degenerate $\delta _{a}$	$e^{ta}$	$e^{ita}$
Bernoulli $P(X=1)=p$	$1-p+pe^{t}$	$1-p+pe^{it}$
Geometric $(1-p)^{k-1}\,p$	${\frac {pe^{t}}{1-(1-p)e^{t}}}$ $\forall t<-\ln(1-p)$	${\frac {pe^{it}}{1-(1-p)\,e^{it}}}$
Binomial $B(n,p)$	$\left(1-p+pe^{t}\right)^{n}$	$\left(1-p+pe^{it}\right)^{n}$
Negative Binomial $NB(r,p)$	${\frac {(1-p)^{r}}{\left(1-pe^{t}\right)^{r}}}$	${\frac {(1-p)^{r}}{\left(1-pe^{it}\right)^{r}}}$
Poisson $Pois(\lambda )$	$e^{\lambda (e^{t}-1)}$	$e^{\lambda (e^{it}-1)}$
Uniform (continuous) $U(a,b)$	${\frac {e^{tb}-e^{ta}}{t(b-a)}}$	${\frac {e^{itb}-e^{ita}}{it(b-a)}}$
Uniform (discrete) $DU(a,b)$	${\frac {e^{at}-e^{(b+1)t}}{(b-a+1)(1-e^{t})}}$	${\frac {e^{ait}-e^{(b+1)it}}{(b-a+1)(1-e^{it})}}$
Laplace $L(\mu ,b)$	${\frac {e^{t\mu }}{1-b^{2}t^{2}}},~\|t\|<1/b$	${\frac {e^{it\mu }}{1+b^{2}t^{2}}}$
Normal $N(\mu ,\sigma ^{2})$	$e^{t\mu +{\frac {1}{2}}\sigma ^{2}t^{2}}$	$e^{it\mu -{\frac {1}{2}}\sigma ^{2}t^{2}}$
Chi-squared $\mathrm {X} _{k}^{2}$	$(1-2t)^{-{\frac {k}{2}}}$	$(1-2it)^{-{\frac {k}{2}}}$
Noncentral chi-squared $\mathrm {X} _{k}^{2}(\lambda )$	$e^{\lambda t/(1-2t)}(1-2t)^{-{\frac {k}{2}}}$	$e^{i\lambda t/(1-2it)}(1-2it)^{-{\frac {k}{2}}}$
Gamma $\Gamma (k,\theta )$	$(1-t\theta )^{-k},~\forall t<{\tfrac {1}{\theta }}$	$(1-it\theta )^{-k}$
Exponential $Exp(\lambda )$	$\left(1-t\lambda ^{-1}\right)^{-1},~t<\lambda$	$\left(1-it\lambda ^{-1}\right)^{-1}$
Multivariate normal $N(\mathbf {\mu } ,\mathbf {\Sigma } )$	$e^{\mathbf {t} ^{\mathrm {T} }\left({\boldsymbol {\mu }}+{\frac {1}{2}}\mathbf {\Sigma t} \right)}$	$e^{\mathbf {t} ^{\mathrm {T} }\left(i{\boldsymbol {\mu }}-{\frac {1}{2}}{\boldsymbol {\Sigma }}\mathbf {t} \right)}$
Cauchy $Cauchy(\mu ,\theta )$	Does not exist	$e^{it\mu -\theta \|t\|}$
Multivariate Cauchy $MultiCauchy(\mu ,\Sigma )$ ^[2]	Does not exist	$\!\,e^{i\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\mu }}-{\sqrt {\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\Sigma }}\mathbf {t} }}}$

Calculation

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

For a discrete probability mass function, $M_{X}(t)=\sum _{i=1}^{\infty }e^{tx_{i}}\,p_{i}$
For a continuous probability density function, $M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}f(x)\,dx$
In the general case: $M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}\,dF(x)$ , using the Riemann–Stieltjes integral, and where $F$ is the cumulative distribution function.

Note that for the case where $X$ has a continuous probability density function $f(x)$ , $M_{X}(-t)$ is the two-sided Laplace transform of $f(x)$ .

{\begin{aligned}M_{X}(t)&=\int _{-\infty }^{\infty }e^{tx}f(x)\,dx\\&=\int _{-\infty }^{\infty }\left(1+tx+{\frac {t^{2}x^{2}}{2!}}+\cdots +{\frac {t^{n}x^{n}}{n!}}+\cdots \right)f(x)\,dx\\&=1+tm_{1}+{\frac {t^{2}m_{2}}{2!}}+\cdots +{\frac {t^{n}m_{n}}{n!}}+\cdots ,\end{aligned}}

where $m_{n}$ is the $n$ th moment.

Linear combination of independent random variables

If $S_{n}=\sum _{i=1}^{n}a_{i}X_{i}$ , where the X_i are independent random variables and the a_i are constants, then the probability density function for S_n is the convolution of the probability density functions of each of the X_i, and the moment-generating function for S_n is given by

M_{S_{n}}(t)=M_{X_{1}}(a_{1}t)M_{X_{2}}(a_{2}t)\cdots M_{X_{n}}(a_{n}t)\,.

Vector-valued random variables

For vector-valued random variables $\mathbf {X}$ with real components, the moment-generating function is given by

M_{X}(\mathbf {t} )=E\left(e^{\langle \mathbf {t} ,\mathbf {X} \rangle }\right)

where $\mathbf {t}$ is a vector and $\langle \cdot ,\cdot \rangle$ is the dot product.

Important properties

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,

M_{X}(t)=M_{Y}(t),\,

then

F_{X}(x)=F_{Y}(x)\,

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

\lim _{n\rightarrow \infty }\sum _{i=0}^{n}{\frac {t^{i}m_{i}}{i!}}

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

Calculations of moments

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:

m_{n}=E\left(X^{n}\right)=M_{X}^{(n)}(0)=\left.{\frac {d^{n}M_{X}}{dt^{n}}}\right|_{t=0}.

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 properties

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

M_{X}(t)\geq e^{\mu t},

where $\mu$ 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:

M_{X}(t)\geq {\frac {t^{k}E[X^{k}]}{k!}}.

The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X. Since $x\mapsto e^{xt}$ is monotonically increasing for $t>0$ , we have

P(X\geq a)=P(e^{tX}\geq e^{ta})\leq e^{-at}E[e^{tX}]=e^{-at}M_{X}(t)

for any $t>0$ and any a, provided $M_{X}(t)$ exists. For example, when X is a standard normal distribution and $a>0$ , we can choose $t=a$ and recall that $M_{X}(t)=e^{t^{2}/2}$ . This gives $P(X\geq a)\leq e^{-a^{2}/2}$ , which is within a factor of 1+a of the exact value.

Relation to other functions

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

Characteristic function: The characteristic function $\varphi _{X}(t)$ is related to the moment-generating function via $\varphi _{X}(t)=M_{iX}(t)=M_{X}(it):$ 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 $G(z)=E\left[z^{X}\right].\,$ This immediately implies that $G(e^{t})=E\left[e^{tX}\right]=M_{X}(t).\,$

References

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

Casella, George; Berger, Roger. Statistical Inference (2nd ed.). pp. 59–68. ISBN 978-0-534-24312-8.

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[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

[Gentle-3] Grimmett, Geoffrey (1986). Probability - An Introduction. Oxford University Press. p. 101. ISBN 978-0-19-853264-4.

Theory of probability distributions
probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function
raw moment central moment mean variance standard deviation skewness kurtosis L-moment
moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant