1.4: General Aspects of Probability Distributions

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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
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\(\newcommand \gtwid{\approx}\)

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\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
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\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ

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Discrete and Continuous Distributions

Consider a system whose possible configurations \(\sket{n}\) can be labeled by a discrete variable \(n\in \CC\), where \(\CC\) is the set of possible configurations. The total number of possible configurations, which is to say the order of the set \(\CC\), may be finite or infinite. Next, consider an ensemble of such systems, and let \(P\ns_n\) denote the probability that a given random element from that ensemble is in the state (configuration) \(\sket{n}\). The collection \(\{P\ns_n\}\) forms a discrete probability distribution. We assume that the distribution is normalized, meaning \[\sum_{n\in\CC} P\ns_n=1\ .\]

Now let \(A\ns_n\) be a quantity which takes values depending on \(n\). The average of \(A\) is given by \[\langle A\rangle =\sum_{n\in\CC} P\ns_n\,A\ns_n\ .\] Typically, \(\CC\) is the set of integers (\(\MZ\)) or some subset thereof, but it could be any countable set. As an example, consider the throw of a single six-sided die. Then \(P\ns_n=\frac{1}{6}\) for each \(n\in\{1,\ldots,6\}\). Let \(A\ns_n=0\) if \(n\) is even and \(1\) if \(n\) is odd. Then find \(\langle A\rangle=\half\), on average half the throws of the die will result in an even number.

It may be that the system’s configurations are described by several discrete variables \(\{n\ns_1,n\ns_2,n\ns_3,\ldots\}\). We can combine these into a vector \(\Bn\) and then we write \(P\ns_\Bn\) for the discrete distribution, with \(\sum_\Bn P\ns_\Bn=1\).

Another possibility is that the system’s configurations are parameterized by a collection of continuous variables, \(\Bvphi=\{\varphi\ns_1,\ldots,\varphi\ns_n\}\). We write \(\Bvphi\in\ROmega\), where \(\ROmega\) is the phase space (or configuration space) of the system. Let \(d\mu\) be a measure on this space. In general, we can write \[d\mu=W(\varphi\ns_1,\ldots,\varphi\ns_n)\, d\varphi\ns_1\,d\varphi\ns_2\cdots d\varphi\ns_n\ .\] The phase space measure used in classical statistical mechanics gives equal weight \(W\) to equal phase space volumes: \[d\mu=\CC\prod_{\sigma=1}^r dq\ns_\sigma\,dp\ns_\sigma\ ,\] where \(\CC\) is a constant we shall discuss later on below⁸.

Any continuous probability distribution \(P(\Bvphi)\) is normalized according to \[\int\limits_\ROmega\!\!d\mu\,P(\Bvphi)=1\ .\] The average of a function \(A(\Bvphi)\) on configuration space is then \[\langle A\rangle =\int\limits_\ROmega\!\!d\mu\, P(\Bvphi)\,A(\Bvphi)\ .\] For example, consider the Gaussian distribution \[P(x)={1\over\sqrt{2\pi\sigma^2}}\,e^{-(x-\mu)^2/2\sigma^2}\ . \label{pgauss}\] From the result⁹ \[\impi dx\>e^{-\alpha x^2}\,e^{-\beta x}=\sqrt{\pi\over\alpha} \ e^{\beta^2/4\alpha}\ ,\] we see that \(P(x)\) is normalized. One can then compute \[\begin{split} \langle x\rangle&=\mu\\ \langle x^2\rangle-\langle x\rangle^2 &=\sigma^2\ . \end{split}\] We call \(\mu\) the mean and \(\sigma\) the standard deviation of the distribution, Equation [pgauss].

The quantity \(P(\Bvphi)\) is called the distribution or probability density. One has \[P(\Bvphi)\,d\mu = \hbox{probability that configuration lies within volume $d\mu$ centered at $\Bvphi$}\] For example, consider the probability density \(P=1\) normalized on the interval \(x\in\big[0,1\big]\). The probability that some \(x\) chosen at random will be exactly \(\half\), say, is infinitesimal – one would have to specify each of the infinitely many digits of \(x\). However, we can say that \(x\in\big[0.45\,,\,0.55\big]\) with probability \(\frac{1}{10}\).

If \(x\) is distributed according to \(P\ns_1(x)\), then the probability distribution on the product space \((x\ns_1\,,\,x\ns_2)\) is simply the product of the distributions: \(P\ns_2(x\ns_1,x\ns_2)=P\ns_1(x\ns_1)\,P\ns_1(x\ns_2)\). Suppose we have a function \(\phi(x\ns_1,\ldots,x\ns_N)\). How is it distributed? Let \(P(\phi)\) be the distribution for \(\phi\). We then have \[\begin{split} P(\phi)&=\impi dx\ns_1\cdots\impi dx\ns_N\,P\ns_N(x\ns_1,\ldots,x\ns_N)\> \delta\Big(\phi(x\ns_1,\ldots,x\ns_N)-\phi\Big)\\ &=\impi dx\ns_1\cdots\!\impi dx\ns_N\,P\ns_1(x\ns_1)\cdots P\ns_1(x\ns_N)\> \delta\Big(\phi(x\ns_1,\ldots,x\ns_N)-\phi\Big)\ , \end{split}\] where the second line is appropriate if the \(\{x\ns_j\}\) are themselves distributed independently. Note that \[\impi d\phi\>P(\phi) = 1\ ,\] so \(P(\phi)\) is itself normalized.

Central limit theorem

In particular, consider the distribution function of the sum \(X=\sum_{i=1}^N x\ns_i\). We will be particularly interested in the case where \(N\) is large. For general \(N\), though, we have \[P\ns_N(X)=\impi dx\ns_1\cdots\!\impi dx\ns_N\,P\ns_1(x\ns_1)\cdots P\ns_1(x\ns_N)\> \delta\big(x\ns_1+x\ns_2+\ldots+x\ns_N-X\big)\ .\] It is convenient to compute the Fourier transform¹⁰ of \(P(X)\): \[\begin{split} {\hat P}\ns_N(k)&=\impi dX\,P\ns_N(X)\,e^{-ikX}\\ &=\impi dX\!\!\impi dx\ns_1\cdots\!\impi dx\ns_N\,P\ns_1(x\ns_1)\cdots P\ns_1(x\ns_N) \>\delta\big(x\ns_1+\ldots+x\ns_N-X)\,e^{-ikX}=\big[{\hat P\ns_1}(k)\big]^N\ , \end{split}\] where \[{\hat P}\ns_1(k)=\impi dx\,P\ns_1(x)\,e^{-ikx}\] is the Fourier transform of the single variable distribution \(P\ns_1(x)\). The distribution \(P\ns_N(X)\) is a convolution of the individual \(P\ns_1(x\ns_i)\) distributions. We have therefore proven that the Fourier transform of a convolution is the product of the Fourier transforms.

OK, now we can write for \({\hat P}\ns_1(k)\) \[\begin{split} {\hat P}\ns_1(k)&=\impi dx\,P\ns_1(x)\,\big(1-ikx -\half \,k^2 x^2 +\frac{1}{6}\,i \,k^3\,x^3 + \ldots\big)\\ &=1-ik\langle x\rangle -\half\, k^2\langle x^2\rangle +\frac{1}{6} \, i\,k^3\langle x^3\rangle + \ldots \ . \end{split}\] Thus, \[\ln{\hat P}\ns_1(k)=-i\mu k -\half\sigma^2 k^2 + \frac{1}{6}\,i\,\gamma^3\,k^3+\ldots\ ,\] where \[\begin{split} \mu&=\langle x\rangle\\ \sigma^2&=\langle x^2\rangle - \langle x\rangle ^2\\ \gamma^3&=\langle x^3\rangle-3\,\langle x^2\rangle\,\langle x\rangle + 2\,\langle x\rangle^3 \end{split}\] We can now write \[\big[{\hat P}\ns_1(k)\big]^N=e^{-iN\mu k}\, e^{-N \sigma^2 k^2/2}\, e^{iN\gamma^3 k^3/6}\cdots\] Now for the inverse transform. In computing \(P\ns_N(X)\), we will expand the term \(e^{iN\gamma^3 k^3/6}\) and all subsequent terms in the above product as a power series in \(k\). We then have \[\begin{split} P\ns_N(X)&=\impi{dk\over 2\pi}\ e^{ik(X-N\mu)}\,e^{-N\sigma^2 k^2/2}\>\Big\{1+\frac{1}{6}\, i\,N\gamma^3 k^3 + \ldots\Big\}\\ &=\bigg(1-{\gamma^3\over 6} N\, {\pz^3\over\pz X^3} + \ldots \bigg) {1\over\sqrt{2\pi N\sigma^2}}\, e^{-(X-N\mu)^2/2N\sigma^2}\bvph\\ &=\bigg(1-{\gamma^3\over 6}\,N^{-1/2}\, {\pz^3\over\pz \xi^3} + \ldots \bigg) {1\over\sqrt{2\pi N\sigma^2}}\, e^{-\xi^2/2\sigma^2}\ . \end{split}\] In going from the second line to the third, we have written \(X=N\mu+\sqrt{N}\,\xi\), in which case \(\pz\ns_X=N^{-1/2}\,\pz\ns_\xi\), and the non-Gaussian terms give a subleading contribution which vanishes in the \(N\to\infty\) limit. We have just proven the central limit theorem: in the limit \(N\to\infty\), the distribution of a sum of \(N\) independent random variables \(x\ns_i\) is a Gaussian with mean \(N\mu\) and standard deviation \(\sqrt{N}\,\sigma\). Our only assumptions are that the mean \(\mu\) and standard deviation \(\sigma\) exist for the distribution \(P\ns_1(x)\). Note that \(P\ns_1(x)\) itself need not be a Gaussian – it could be a very peculiar distribution indeed, but so long as its first and second moment exist, where the \(k^\ssr{th}\) moment is simply \(\langle x^k\rangle\), the distribution of the sum \(X=\sum_{i=1}^N x\ns_i\) is a Gaussian.

Moments and cumulants

Consider a general multivariate distribution \(P(x\ns_1,\ldots,x\ns_N)\) and define the multivariate Fourier transform \[\HP(k\ns_1,\ldots,k\ns_N)=\impi dx\ns_1\cdots\!\!\impi dx\ns_N\>P(x\ns_1,\ldots,x\ns_N)\,\exp\bigg(\!-i\sum_{j=1}^N k\ns_j x\ns_j\bigg)\ .\] The inverse relation is \[P(x\ns_1,\ldots,x\ns_N)=\impi {dk\ns_1\over 2\pi}\cdots\!\!\impi {dk\ns_N\over 2\pi}\> \HP(k\ns_1,\ldots,k\ns_N)\,\exp\bigg(\!+i\sum_{j=1}^N k\ns_j x\ns_j\bigg)\ .\] Acting on \(\HP(\Bk)\), the differential operator \(i\,{\pz\over\pz k\ns_i}\) brings down from the exponential a factor of \(x\ns_i\) inside the integral. Thus, \[\Bigg[\bigg(\!i\,{\pz\over\pz k\ns_1}\bigg)^{\!m\ns_1}\!\!\cdots\bigg(\!i\,{\pz\over\pz k\ns_N}\bigg)^{\!m\ns_N}\,\HP(\Bk) \Bigg]\nd_{\Bk=0} \!\!\!\!=\blangle x_1^{m\ns_1}\cdots x_N^{m\ns_N}\brangle\ .\] Similarly, we can reconstruct the distribution from its moments, viz. \[\HP(\Bk)=\sum_{m\ns_1=0}^\infty\cdots\sum_{m\ns_N=0}^\infty {(-i k\ns_1)^{m\ns_1}\over m\ns_1!}\cdots {(-i k\ns_N)^{m\ns_N}\over m\ns_N!}\, \blangle x_1^{m\ns_1}\cdots x_N^{m\ns_N}\brangle\ .\]

The cumulants \(\langle\!\langle x_1^{m\ns_1}\cdots x_N^{m\ns_N} \rangle\!\rangle\) are defined by the Taylor expansion of \(\ln \HP(\Bk)\): \[\ln\HP(\Bk)=\sum_{m\ns_1=0}^\infty\cdots\sum_{m\ns_N=0}^\infty {(-i k\ns_1)^{m\ns_1}\over m\ns_1!}\cdots {(-i k\ns_N)^{m\ns_N}\over m\ns_N!}\, \big\langle\!\big\langle x_1^{m\ns_1}\cdots x_N^{m\ns_N}\big\rangle\!\big\rangle\ .\] There is no general form for the cumulants. It is straightforward to derive the following low order results: \[\begin{split} \langle\!\langle x\ns_i \rangle\!\rangle&= \langle x\ns_i \rangle \\ \langle\!\langle x\ns_i x\ns_j\rangle\!\rangle&= \langle x\ns_i x\ns_j \rangle - \langle x\ns_i\rangle \langle x\ns_j\rangle \\ \langle\!\langle x\ns_i x\ns_j x\ns_k \rangle\!\rangle&= \langle x\ns_i x\ns_j x\ns_k \rangle - \langle x\ns_i x\ns_j \rangle \langle x\ns_k \rangle - \langle x\ns_j x\ns_k \rangle \langle x\ns_i \rangle - \langle x\ns_k x\ns_i \rangle \langle x\ns_j \rangle + 2 \langle x\ns_i \rangle \langle x\ns_j \rangle \langle x\ns_k \rangle \ . \end{split}\]

Multidimensional Gaussian integral

Consider the multivariable Gaussian distribution, \[P(\Bx)\equiv \bigg({\det\!A\over (2\pi)^n}\bigg)^{\!1/2}\exp\Big(-\half \, x\ns_i \, A\ns_{ij} \, x\ns_j\Big)\ ,\] where \(A\) is a positive definite matrix of rank \(n\). A mathematical result which is extremely important throughout physics is the following: \[Z(\Bb)=\bigg({\det\!A\over (2\pi)^n}\bigg)^{\!1/2}\impi dx\ns_1\cdots\!\impi dx\ns_n \> \exp\Big(-\half \, x\ns_i \, A\ns_{ij} \, x\ns_j + b\ns_i\,x\ns_i\Big)=\exp\Big(\half \, b\ns_i\,A^{-1}_{ij}\,b\ns_j\Big)\ .\] Here, the vector \(\Bb=(b\ns_1\,,\,\ldots\,,\,b\ns_n)\) is identified as a source. Since \(Z(0)=1\), we have that the distribution \(P(\Bx)\) is normalized. Now consider averages of the form \[\begin{split} \langle \, x\ns_{j\ns_1}\!\!\cdots \,x\ns_{j\ns_{2k}} \, \rangle &= \int\!d^n\!x\>P(\Bx)\>x\ns_{j\ns_1}\!\!\cdots \,x\ns_{j\ns_{2k}} ={\pz^n\!Z(\Bb)\over\pz b\ns_{j\ns_1}\!\cdots\>\pz b\ns_{j\ns_{2k}}}\bigg|\nd_{\Bb=0} \\ &=\sum_{contractions} \!\!\! A^{-1}_{j\ns_{\sigma(1)} j\ns_{\sigma(2)}} \!\!\cdots A^{-1}_{j\ns_{\sigma(2k-1)} j\ns_{\sigma(2k)}}\ . \end{split}\] The sum in the last term is over all contractions of the indices \(\{j\ns_1\,,\,\ldots\,,\,j\ns_{2k}\}\). A contraction is an arrangement of the \(2k\) indices into \(k\) pairs. There are \(C\ns_{2k}=(2k)!/2^k k!\) possible such contractions. To obtain this result for \(C\ns_k\), we start with the first index and then find a mate among the remaining \(2k-1\) indices. Then we choose the next unpaired index and find a mate among the remaining \(2k-3\) indices. Proceeding in this manner, we have \[C\ns_{2k}=(2k-1)\cdot(2k-3)\cdots 3\cdot 1 = {(2k)!\over 2^k k!}\ .\] Equivalently, we can take all possible permutations of the \(2k\) indices, and then divide by \(2^k k!\) since permutation within a given pair results in the same contraction and permutation among the \(k\) pairs results in the same contraction. For example, for \(k=2\), we have \(C\ns_4=3\), and \[\langle \, x\ns_{j\ns_1} x\ns_{j\ns_2} x\ns_{j\ns_3} x\ns_{j\ns_4 \, }\rangle = A^{-1}_{j\ns_1 j\ns_2} A^{-1}_{j\ns_3 j\ns_4} + A^{-1}_{j\ns_1 j\ns_3} A^{-1}_{j\ns_2 j\ns_4} + A^{-1}_{j\ns_1 j\ns_4} A^{-1}_{j\ns_2 j\ns_3} \ .\] If we define \(b\ns_i=ik\ns_i\), we have \[\HP(\Bk)=\exp\Big(\!-\half\, k\ns_i \, A^{-1}_{ij}\,k\ns_j\Big)\ ,\] from which we read off the cumulants \(\langle\!\langle x\ns_i x\ns_j \rangle\!\rangle= A^{-1}_{ij}\), with all higher order cumulants vanishing.