Moment-generating function

Template:Short description 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 Template:Mvar-th moment about 0 is the Template:Mvar-th derivative of the moment-generating function, evaluated at 0.

In addition to univariate real-valued distributions, moment-generating functions can also 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.

DefinitionEdit

Let <math> X </math> be a random variable with CDF <math>F_X</math>. The moment generating function (mgf) of <math>X</math> (or <math>F_X</math>), denoted by <math>M_X(t)</math>, is

<math display="block"> M_X(t) = \operatorname E \left[e^{tX}\right] </math>

provided this expectation exists for <math>t</math> in some open neighborhood of 0. That is, there is an <math>h > 0</math> such that for all <math>t</math> in <math>-h < 0 < h</math>, <math>\operatorname E \left[e^{tX}\right] </math> exists. If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist.<ref>Template:Cite book</ref>

In other words, the moment-generating function of Template:Mvar is the expectation of the random variable <math> e^{tX}</math>. More generally, when <math>\mathbf X = ( X_1, \ldots, X_n)^{\mathrm{T}}</math>, an <math>n</math>-dimensional random vector, and <math>\mathbf t</math> is a fixed vector, one uses <math>\mathbf t \cdot \mathbf X = \mathbf t^\mathrm T\mathbf X</math> instead of Template:Nowrap <math display="block"> M_{\mathbf X}(\mathbf t) := \operatorname E \left[e^{\mathbf t^\mathrm T\mathbf X}\right].</math>

<math> M_X(0) </math> 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.<ref>Template:Cite book</ref> The series expansion of <math>e^{tX}</math> is

<math display="block"> 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. </math>

Hence, <math display="block">\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}</math>

where <math>m_n</math> is the Template:Nowrap moment. Differentiating <math>M_X(t)</math> <math>i</math> times with respect to <math>t</math> and setting <math>t = 0</math>, we obtain the <math>i</math>-th moment about the origin, <math>m_i</math>; see Template:Slink below.

If <math>X</math> is a continuous random variable, the following relation between its moment-generating function <math>M_X(t)</math> and the two-sided Laplace transform of its probability density function <math>f_X(x)</math> holds:

<math display="block">M_X(t) = \mathcal{L}\{f_X\}(-t),</math>

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

<math display="block">\mathcal{L}\{f_X\}(s) = \int_{-\infty}^\infty e^{-sx} f_X(x)\, dx,</math>

and the moment-generating function's definition expands (by the law of the unconscious statistician) to <math display="block">M_X(t) = \operatorname E \left[e^{tX}\right] = \int_{-\infty}^\infty e^{tx} f_X(x)\, dx.</math>

This is consistent with the characteristic function of <math>X</math> being a Wick rotation of <math>M_X(t)</math> when the moment generating function exists, as the characteristic function of a continuous random variable <math>X</math> is the Fourier transform of its probability density function <math>f_X(x)</math>, and in general when a function <math>f(x)</math> is of exponential order, the Fourier transform of <math>f</math> 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 <math>M_X(t)</math> when the latter exists.

Distribution	Moment-generating function <math>M_X(t)</math>	Characteristic function <math>\varphi (t)</math>
Degenerate <math>\delta_a</math>	<math>e^{ta}</math>	<math>e^{ita}</math>
Bernoulli <math>P(X = 1) = p</math>	<math>1 - p + pe^t</math>	<math>1 - p + pe^{it}</math>
Binomial <math>B(n, p)</math>	<math>\left(1 - p + pe^t\right)^n</math>	<math>\left(1 - p + pe^{it}\right)^n</math>
Geometric <math>(1 - p)^{k}\,p</math>	<math>\frac{p}{1 - (1 - p) e^t}, ~ t < -\ln(1 - p)</math>	<math>\frac{p}{1 - (1 - p)\,e^{it}}</math>
Negative binomial <math>\operatorname{NB}(r, p)</math>	<math>\left(\frac{p}{1 - e^t + pe^t}\right)^r, ~ t<-\ln(1-p)</math>	<math>\left(\frac{p}{1 - e^{it} + pe^{it}}\right)^r</math>
Poisson <math>\operatorname{Pois}(\lambda)</math>	<math>e^{\lambda(e^t - 1)}</math>	<math>e^{\lambda(e^{it} - 1)}</math>
Uniform (continuous) <math>\operatorname U(a, b)</math>	<math>\frac{e^{tb} - e^{ta}}{t(b - a)}</math>	<math>\frac{e^{itb} - e^{ita}}{it(b - a)}</math>
Uniform (discrete) <math>\operatorname{DU}(a, b)</math>	<math>\frac{e^{at} - e^{(b + 1)t}}{(b - a + 1)(1 - e^t)}</math>	<math>\frac{e^{ait} - e^{(b + 1)it}}{(b - a + 1)(1 - e^{it})}</math>
Laplace <math>L(\mu, b)</math>	t\| < 1/b</math>	<math>\frac{e^{it\mu}}{1 + b^2t^2}</math>
Normal <math>N(\mu, \sigma^2)</math>	<math>e^{t\mu + \sigma^2 t^2 / 2}</math>	<math>e^{it\mu - \sigma^2 t^2 / 2}</math>
Chi-squared <math>\chi^2_k</math>	<math>{\left(1 - 2t\right)}^{-k/2}, ~ t < 1/2</math>	<math>{\left(1 - 2it\right)}^{-{k}/{2}}</math>
Noncentral chi-squared <math>\chi^2_k(\lambda)</math>	<math>e^{\lambda t/(1-2t)} {\left(1 - 2t\right)}^{-k/2}</math>	<math>e^{i\lambda t/(1-2it)} {\left(1 - 2it\right)}^{-k/2}</math>
Gamma <math>\Gamma(k, \tfrac{1}{\theta})</math>	<math>{\left(1 - t\theta\right)}^{-k}, ~ t < \tfrac{1}{\theta}</math>	<math>{\left(1 - it\theta\right)}^{-k}</math>
Exponential <math>\operatorname{Exp}(\lambda)</math>	<math>\left(1 - t\lambda^{-1}\right)^{-1}, ~ t < \lambda</math>	<math>\left(1 - it\lambda^{-1}\right)^{-1}</math>
Beta	<math>1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}</math>	<math>{}_1F_1(\alpha; \alpha+\beta; i\,t)\! </math> (see Confluent hypergeometric function)
Multivariate normal <math>N(\mathbf{\mu}, \mathbf{\Sigma})</math>	<math>\exp\left[\mathbf{t}^\mathrm{T} \left( \boldsymbol{\mu} + \tfrac{1}{2} \boldsymbol{\Sigma} \mathbf{t}\right)\right]</math>	<math>\exp\left[\mathbf{t}^\mathrm{T} \left(i \boldsymbol{\mu} - \tfrac{1}{2} \boldsymbol{\Sigma} \mathbf{t}\right)\right]</math>
Cauchy <math>\operatorname{Cauchy}(\mu, \theta)</math>	Does not exist	t\|}</math>
Multivariate Cauchy <math>\operatorname{MultiCauchy}(\mu, \Sigma)</math><ref>Kotz et al.Template:Full citation needed p. 37 using 1 as the number of degree of freedom to recover the Cauchy distribution</ref>	Does not exist	<math>\exp\left(i\mathbf{t}^{\mathrm{T}}\boldsymbol\mu - \sqrt{\mathbf{t}^{\mathrm{T}}\boldsymbol{\Sigma} \mathbf{t}}\right)</math>

CalculationEdit

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, <math>M_X(t)=\sum_{i=0}^\infty e^{tx_i}\, p_i</math>
For a continuous probability density function, <math> M_X(t) = \int_{-\infty}^\infty e^{tx} f(x)\,dx </math>
In the general case: <math>M_X(t) = \int_{-\infty}^\infty e^{tx}\,dF(x)</math>, using the Riemann–Stieltjes integral, and where <math>F</math> is the cumulative distribution function. This is simply the Laplace-Stieltjes transform of <math>F</math>, but with the sign of the argument reversed.

Note that for the case where <math>X</math> has a continuous probability density function <math>f(x)</math>, <math>M_X(-t)</math> is the two-sided Laplace transform of <math>f(x)</math>.

<math display="block">\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}</math>

where <math>m_n</math> is the <math>n</math>th moment.

Linear transformations of random variablesEdit

If random variable <math>X</math> has moment generating function <math>M_X(t)</math>, then <math>\alpha X + \beta</math> has moment generating function <math>M_{\alpha X + \beta}(t) = e^{\beta t}M_X(\alpha t)</math>

<math display="block"> 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) </math>

Linear combination of independent random variablesEdit

If <math display="inline">S_n = \sum_{i=1}^n a_i X_i</math>, where the Template:Math are independent random variables and the Template:Math are constants, then the probability density function for Template:Math is the convolution of the probability density functions of each of the Template:Math, and the moment-generating function for Template:Math is given by

<math display="block"> M_{S_n}(t) = M_{X_1}(a_1t) M_{X_2}(a_2t) \cdots M_{X_n}(a_nt) \, . </math>

Vector-valued random variablesEdit

For vector-valued random variables <math>\mathbf X</math> with real components, the moment-generating function is given by

<math display="block"> M_X(\mathbf t) = \operatorname{E}\left[e^{\langle \mathbf t, \mathbf X \rangle}\right] </math>

where <math>\mathbf t</math> is a vector and <math>\langle \cdot, \cdot \rangle</math> is the dot product.

Important propertiesEdit

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

An important property of the moment-generating function is that it uniquely determines the distribution. In other words, if <math>X</math> and <math>Y</math> are two random variables and for all values of Template:Mvar,

for all values of Template:Mvar (or equivalently Template:Mvar and Template:Mvar 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

<math display="block">\lim_{n \to \infty} \sum_{i=0}^n \frac{t^i m_i}{i!}</math>

may not exist. The log-normal 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 Template:Math, then it is the exponential generating function of the moments of the probability distribution:

<math display="block">m_n = \operatorname{E}\left[ X^n \right] = M_X^{(n)}(0) = \left. \frac{d^n M_X}{dt^n}\right|_{t=0}.</math>

That is, with Template:Mvar being a nonnegative integer, the Template:Mvar-th moment about 0 is the Template:Mvar-th derivative of the moment generating function, evaluated at Template:Math.

Other propertiesEdit

Jensen's inequality provides a simple lower bound on the moment-generating function: <math display="block"> M_X(t) \geq e^{\mu t}, </math> where <math>\mu</math> is the mean of Template:Mvar.

The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable Template:Mvar. This statement is also called the Chernoff bound. Since <math>x \mapsto e^{xt}</math> is monotonically increasing for <math>t>0</math>, we have <math display="block"> \Pr(X \ge a) = \Pr(e^{tX} \ge e^{ta}) \le e^{-at} \operatorname{E}\left[e^{tX}\right] = e^{-at}M_X(t)</math> for any <math>t>0</math> and any Template:Mvar, provided <math>M_X(t)</math> exists. For example, when Template:Mvar is a standard normal distribution and <math>a > 0</math>, we can choose <math>t=a</math> and recall that <math>M_X(t)=e^{t^2/2}</math>. This gives <math>\Pr(X\ge a)\le e^{-a^2/2}</math>, which is within a factor of Template:Math 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 <math>X</math> is non-negative, the moment generating function gives a simple, useful bound on the moments: <math display="block">\operatorname{E}[X^m] \le \left(\frac{m}{te}\right)^m M_X(t),</math> For any <math>X,m\ge 0</math> and <math>t>0</math>.

This follows from the inequality <math>1+x\le e^x</math> into which we can substitute <math>x'=tx/m-1</math> implies <math>tx/m\le e^{tx/m-1}</math> for any Template:Nowrap Now, if <math>t > 0</math> and <math>x,m\ge 0</math>, this can be rearranged to <math>x^m \le (m/(te))^m e^{tx}</math>. Taking the expectation on both sides gives the bound on <math>\operatorname{E}[X^m]</math> in terms of <math>\operatorname{E}[e^{tX}]</math>.

As an example, consider <math>X\sim\text{Chi-Squared}</math> with <math>k</math> degrees of freedom. Then from the examples <math>M_X(t) = (1-2t)^{-k/2}</math>. Picking <math>t=m/(2m+k)</math> and substituting into the bound: <math display="block">\operatorname{E}[X^m] \le {\left(1 + 2m/k\right)}^{k/2} e^{-m} {\left(k + 2m\right)}^m.</math> We know that in this case the correct bound is <math>\operatorname{E}[X^m]\le 2^m \Gamma(m+k/2)/\Gamma(k/2)</math>. To compare the bounds, we can consider the asymptotics for large <math>k</math>. Here the moment-generating function bound is <math>k^m(1+m^2/k + O(1/k^2))</math>, where the real bound is <math>k^m(1+(m^2-m)/k + O(1/k^2))</math>. The moment-generating function bound is thus very strong in this case.

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 <math>\varphi_X(t)</math> is related to the moment-generating function via <math>\varphi_X(t) = M_{iX}(t) = M_X(it):</math> 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 <math>G(z) = \operatorname{E}\left[z^X\right].</math> This immediately implies that <math>G(e^t) = \operatorname{E}\left[e^{tX}\right] = M_X(t).</math>