Moment-generating function

Let $X$ be a random variable with CDF $F\_X$. The moment generating function (mgf) of $X$ (or $F\_X$), denoted by $M\_X(t)$, is

$$M\_X(t)\; =\; \backslash operatorname\; E\; \backslash left[e^\{tX\}\backslash right]$$

provided this expectation exists for $t$ in some open neighborhood of 0. That is, there is an $h\; >\; 0$ such that for all $t$ in , $\backslash operatorname\; E\; \backslash left[e^\{tX\}\backslash right]$ exists. If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist.{{cite book |last1=Casella |first1=George|last2= Berger|first2= Roger L. |title=Statistical Inference |publisher=Wadsworth & Brooks/Cole|year=1990 |page=61 |isbn=0-534-11958-1 }}

In other words, the moment-generating function of {{mvar|X}} is the expectation of the random variable $e^\{tX\}$. More generally, when $\backslash mathbf\; X\; =\; (\; X\_1,\; \backslash ldots,\; X\_n)^\{\backslash mathrm\{T\}\}$, an $n$-dimensional random vector, and $\backslash mathbf\; t$ is a fixed vector, one uses $\backslash mathbf\; t\; \backslash cdot\; \backslash mathbf\; X\; =\; \backslash mathbf\; t^\backslash mathrm\; T\backslash mathbf\; X$ instead of {{nowrap|$tX$:}}

$$M\_\{\backslash mathbf\; X\}(\backslash mathbf\; t)\; :=\; \backslash operatorname\; E\; \backslash left[e^\{\backslash mathbf\; t^\backslash mathrm\; T\backslash mathbf\; X\}\backslash right].$$

$M\_X(0)$ 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.{{cite book |last=Bulmer |first=M. G. |title=Principles of Statistics |publisher=Dover |year=1979 |pages=75–79 |isbn=0-486-63760-3 }} 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,

$$\backslash begin\{align\}$$

M_X(t) &= \operatorname E [e^{t X}] \\[1ex]

&= 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 \\[1ex]

& = 1 + t m_1 + \frac{t^2 m_2}{2!} + \frac{t^3 m_3}{3!} + \cdots + \frac{t^n m_n}{n!} + \cdots,

\end{align}

where $m\_n$ is the {{nowrap|$n$-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 {{slink|#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:

$$M\_X(t)\; =\; \backslash mathcal\{L\}\backslash \{f\_X\backslash \}(-t),$$

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

$$\backslash mathcal\{L\}\backslash \{f\_X\backslash \}(s)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-sx\}\; f\_X(x)\backslash ,\; dx,$$

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

$$M\_X(t)\; =\; \backslash operatorname\; E\; \backslash left[e^\{tX\}\backslash right]\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{tx\}\; f\_X(x)\backslash ,\; dx.$$

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.

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.

|-

|Multivariate Cauchy

$\backslash operatorname\{MultiCauchy\}(\backslash mu,\; \backslash Sigma)$Kotz et al.{{full citation needed|date=December 2019}} p. 37 using 1 as the number of degree of freedom to recover the Cauchy distribution

|Does not exist

|$\backslash exp\backslash left(i\backslash mathbf\{t\}^\{\backslash mathrm\{T\}\}\backslash boldsymbol\backslash mu\; -\; \backslash sqrt\{\backslash mathbf\{t\}^\{\backslash mathrm\{T\}\}\backslash boldsymbol\{\backslash Sigma\}\; \backslash mathbf\{t\}\}\backslash right)$

|}

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, $M\_X(t)=\backslash sum\_\{i=0\}^\backslash infty\; e^\{tx\_i\}\backslash ,\; p\_i$
• For a continuous probability density function, $M\_X(t)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{tx\}\; f(x)\backslash ,dx$
• In the general case: $M\_X(t)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{tx\}\backslash ,dF(x)$, using the Riemann–Stieltjes integral, and where $F$ is the cumulative distribution function. This is simply the Laplace-Stieltjes transform of $F$, but with the sign of the argument reversed.

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 $f(x)$.

$$\backslash begin\{align\}$$

M_X(t) & = \int_{-\infty}^\infty e^{tx} f(x)\,dx \\[1ex]

& = \int_{-\infty}^\infty \left( 1+ tx + \frac{t^2 x^2}{2!} + \cdots + \frac{t^n x^n}{n!} + \cdots\right) f(x)\,dx \\[1ex]

& = 1 + tm_1 + \frac{t^2 m_2}{2!} + \cdots + \frac{t^n m_n}{n!} +\cdots,

\end{align}

where $m\_n$ is the $n$th moment.

If random variable $X$ has moment generating function $M\_X(t)$, then $\backslash alpha\; X\; +\; \backslash beta$ has moment generating function $M\_\{\backslash alpha\; X\; +\; \backslash beta\}(t)\; =\; e^\{\backslash beta\; t\}M\_X(\backslash alpha\; t)$

M_{\alpha X + \beta}(t) = \operatorname{E}\left[e^{(\alpha X + \beta) t}\right] = e^{\beta t} \operatorname{E}\left[e^{\alpha Xt}\right] = e^{\beta t} M_X(\alpha t)

If $S\_n\; =\; \backslash sum\_\{i=1\}^n\; a\_i\; X\_i$, where the {{math|X_i}} are independent random variables and the {{math|a_i}} are constants, then the probability density function for {{math|S_n}} is the convolution of the probability density functions of each of the {{math|X_i}}, and the moment-generating function for {{math|S_n}} is given by

M_{S_n}(t) = M_{X_1}(a_1t) M_{X_2}(a_2t) \cdots M_{X_n}(a_nt) \, .

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

$$M\_X(\backslash mathbf\; t)\; =\; \backslash operatorname\{E\}\backslash left[e^\{\backslash langle\; \backslash mathbf\; t,\; \backslash mathbf\; X\; \backslash rangle\}\backslash right]$$

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

Moment generating functions are positive and log-convex,{{Citation needed|reason=log-convexity|date=June 2023}} with M(0) = 1.

An important property of the moment-generating function is that it uniquely determines the distribution. In other words, if $X$ and $Y$ are two random variables and for all values of {{mvar|t}},

$$M\_X(t)\; =\; M\_Y(t),$$

then

$$F\_X(x)\; =\; F\_Y(x)$$

for all values of {{mvar|x}} (or equivalently {{mvar|X}} and {{mvar|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

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash sum\_\{i=0\}^n\; \backslash frac\{t^i\; m\_i\}\{i!\}$$

may not exist. The log-normal distribution is an example of when this occurs.

The moment-generating function is so called because if it exists on an open interval around {{math|1=t = 0}}, then it is the exponential generating function of the moments of the probability distribution:

$$m\_n\; =\; \backslash operatorname\{E\}\backslash left[\; X^n\; \backslash right]\; =\; M\_X^\{(n)\}(0)\; =\; \backslash left.\; \backslash frac\{d^n\; M\_X\}\{dt^n\}\backslash right|\_\{t=0\}.$$

That is, with {{mvar|n}} being a nonnegative integer, the {{mvar|n}}-th moment about 0 is the {{mvar|n}}-th derivative of the moment generating function, evaluated at {{math|1=t = 0}}.

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

$$M\_X(t)\; \backslash geq\; e^\{\backslash mu\; t\},$$

where $\backslash mu$ is the mean of {{mvar|X}}.

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

$$\backslash Pr(X\; \backslash ge\; a)\; =\; \backslash Pr(e^\{tX\}\; \backslash ge\; e^\{ta\})\; \backslash le\; e^\{-at\}\; \backslash operatorname\{E\}\backslash left[e^\{tX\}\backslash right]\; =\; e^\{-at\}M\_X(t)$$

for any $t>0$ and any {{mvar|a}}, provided $M\_X(t)$ exists. For example, when {{mvar|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 $\backslash Pr(X\backslash ge\; a)\backslash le\; e^\{-a^2/2\}$, which is within a factor of {{math|1+a}} of the exact value.

Various lemmas, such as Hoeffding's lemma or Bennett's inequality provide bounds on the moment-generating function in the case of a zero-mean, bounded random variable.

When $X$ is non-negative, the moment generating function gives a simple, useful bound on the moments:

$$\backslash operatorname\{E\}[X^m]\; \backslash le\; \backslash left(\backslash frac\{m\}\{te\}\backslash right)^m\; M\_X(t),$$

For any $X,m\backslash ge\; 0$ and $t>0$.

This follows from the inequality $1+x\backslash le\; e^x$ into which we can substitute $x\text{'}=tx/m-1$ implies $tx/m\backslash le\; e^\{tx/m-1\}$ for any {{nowrap|$x,\; t,\; m\; \backslash in\; \backslash mathbb\; R$.}}

Now, if $t\; >\; 0$ and $x,m\backslash ge\; 0$, this can be rearranged to $x^m\; \backslash le\; (m/(te))^m\; e^\{tx\}$.

Taking the expectation on both sides gives the bound on $\backslash operatorname\{E\}[X^m]$ in terms of $\backslash operatorname\{E\}[e^\{tX\}]$.

As an example, consider $X\backslash sim\backslash text\{Chi-Squared\}$ with $k$ degrees of freedom. Then from the examples $M\_X(t)\; =\; (1-2t)^\{-k/2\}$.

Picking $t=m/(2m+k)$ and substituting into the bound:

$$\backslash operatorname\{E\}[X^m]\; \backslash le\; \{\backslash left(1\; +\; 2m/k\backslash right)\}^\{k/2\}\; e^\{-m\}\; \{\backslash left(k\; +\; 2m\backslash right)\}^m.$$

We know that in this case the correct bound is $\backslash operatorname\{E\}[X^m]\backslash le\; 2^m\; \backslash Gamma(m+k/2)/\backslash Gamma(k/2)$.

To compare the bounds, we can consider the asymptotics for large $k$.

Here the moment-generating function bound is $k^m(1+m^2/k\; +\; O(1/k^2))$,

where the real bound is $k^m(1+(m^2-m)/k\; +\; O(1/k^2))$.

The moment-generating function bound is thus very strong in this case.

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

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

• Characteristic function (probability theory)
• Factorial moment generating function
• Rate function
• Hamburger moment problem

{{More footnotes|date=February 2010}}

{{Reflist}}

{{Refbegin}}

• {{cite book |last1=Casella |first1=George |last2=Berger |first2=Roger |title=Statistical Inference |year=2002 |edition=2nd |isbn = 978-0-534-24312-8 |pages=59–68 |publisher=Thomson Learning }}

{{Refend}}

{{Clear}}

{{Theory of probability distributions}}

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Category:Moments (mathematics)

Category:Generating functions