From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Tue, 20 Dec 2022 20:11:25 +0100 Subject: More improvements to knowledge base --- source/know/concept/central-limit-theorem/index.md | 90 +++++++++++++--------- 1 file changed, 52 insertions(+), 38 deletions(-) (limited to 'source/know/concept/central-limit-theorem/index.md') diff --git a/source/know/concept/central-limit-theorem/index.md b/source/know/concept/central-limit-theorem/index.md index 595cee7..e933ee7 100644 --- a/source/know/concept/central-limit-theorem/index.md +++ b/source/know/concept/central-limit-theorem/index.md @@ -18,24 +18,24 @@ the resulting means $$\mu_m$$ are normally distributed across the $$M$$ samples if $$N$$ is sufficiently large. More formally, for $$N$$ independent variables $$x_n$$ with probability distributions $$p(x_n)$$, -the central limit theorem states the following, -where we define the sum $$S$$: +we define the following totals of all variables, means and variances: $$\begin{aligned} - S = \sum_{n = 1}^N x_n - \qquad - \mu_S = \sum_{n = 1}^N \mu_n - \qquad - \sigma_S^2 = \sum_{n = 1}^N \sigma_n^2 + t \equiv \sum_{n = 1}^N x_n + \qquad \qquad + \mu_t \equiv \sum_{n = 1}^N \mu_n + \qquad \qquad + \sigma_t^2 \equiv \sum_{n = 1}^N \sigma_n^2 \end{aligned}$$ -And crucially, it states that the probability distribution $$p_N(S)$$ of $$S$$ for $$N$$ variables +The central limit theorem then states that +the probability distribution $$p_N(t)$$ of $$t$$ for $$N$$ variables will become a normal distribution when $$N$$ goes to infinity: $$\begin{aligned} \boxed{ - \lim_{N \to \infty} \!\big(p_N(S)\big) - = \frac{1}{\sigma_S \sqrt{2 \pi}} \exp\!\Big( -\frac{(\mu_S - S)^2}{2 \sigma_S^2} \Big) + \lim_{N \to \infty} \!\big(p_N(t)\big) + = \frac{1}{\sigma_t \sqrt{2 \pi}} \exp\!\bigg( -\frac{(t - \mu_t)^2}{2 \sigma_t^2} \bigg) } \end{aligned}$$ @@ -45,7 +45,8 @@ Given a probability density $$p(x)$$, its [Fourier transform](/know/concept/four is called the **characteristic function** $$\phi(k)$$: $$\begin{aligned} - \phi(k) = \int_{-\infty}^\infty p(x) \exp(i k x) \dd{x} + \phi(k) + \equiv \int_{-\infty}^\infty p(x) \exp(i k x) \dd{x} \end{aligned}$$ Note that $$\phi(k)$$ can be interpreted as the average of $$\exp(i k x)$$. @@ -54,17 +55,19 @@ where an overline denotes the mean: $$\begin{aligned} \phi(k) - = \sum_{n = 0}^\infty \frac{k^n}{n!} \: \phi^{(n)}(0) - \qquad + = \sum_{n = 0}^\infty \frac{k^n}{n!} \bigg( \dvn{n}{\phi}{k} \Big|_{k = 0} \bigg) + \qquad \qquad \phi(k) - = \overline{\exp(i k x)} = \sum_{n = 0}^\infty \frac{(ik)^n}{n!} \overline{x^n} + = \overline{\exp(i k x)} + = \sum_{n = 0}^\infty \frac{(ik)^n}{n!} \overline{x^n} \end{aligned}$$ By comparing the coefficients of these two power series, we get a useful relation: $$\begin{aligned} - \phi^{(n)}(0) = i^n \: \overline{x^n} + \dvn{n}{\phi}{k} \Big|_{k = 0} + = i^n \: \overline{x^n} \end{aligned}$$ Next, the **cumulants** $$C^{(n)}$$ are defined from the Taylor expansion of $$\ln\!\big(\phi(k)\big)$$: @@ -73,73 +76,82 @@ $$\begin{aligned} \ln\!\big( \phi(k) \big) = \sum_{n = 1}^\infty \frac{(ik)^n}{n!} C^{(n)} \quad \mathrm{where} \quad - C^{(n)} = \frac{1}{i^n} \: \dvn{n}{}{k} \Big(\ln\!\big(\phi(k)\big)\Big) \Big|_{k = 0} + C^{(n)} + \equiv \frac{1}{i^n} \: \dvn{n}{}{k} \ln\!\big(\phi(k)\big) \Big|_{k = 0} \end{aligned}$$ The first two cumulants $$C^{(1)}$$ and $$C^{(2)}$$ are of particular interest, -since they turn out to be the mean and the variance respectively, -using our earlier relation: +since they turn out to be the mean and the variance respectively. +Using our earlier relation: $$\begin{aligned} C^{(1)} - &= - i \dv{}{k} \Big(\ln\!\big(\phi(k)\big)\Big) \Big|_{k = 0} + &= - i \dv{}{k} \ln\!\big(\phi(k)\big) \Big|_{k = 0} = - i \frac{\phi'(0)}{\exp(0)} = \overline{x} \\ C^{(2)} - &= - \dvn{2}{}{k} \Big(\ln\!\big(\phi(k)\big)\Big) \Big|_{k = 0} + &= - \dvn{2}{}{k} \ln\!\big(\phi(k)\big) \Big|_{k = 0} = \frac{\big(\phi'(0)\big)^2}{\exp(0)^2} - \frac{\phi''(0)}{\exp(0)} = - \overline{x}^2 + \overline{x^2} = \sigma^2 \end{aligned}$$ -Let us now define $$S$$ as the sum of $$N$$ independent variables $$x_n$$, in other words: +Now that we have introduced these tools, +we define $$t$$ as the sum +of $$N$$ independent variables $$x_n$$, in other words: $$\begin{aligned} - S = \sum_{n = 1}^N x_n = x_1 + x_2 + ... + x_N + t + \equiv \sum_{n = 1}^N x_n = x_1 + x_2 + ... + x_N \end{aligned}$$ -The probability density of $$S$$ is then as follows, where $$p(x_n)$$ are +The probability density of $$t$$ is then as follows, where $$p(x_n)$$ are the densities of all the individual variables and $$\delta$$ is the [Dirac delta function](/know/concept/dirac-delta-function/): $$\begin{aligned} - p(S) - &= \int\cdots\int_{-\infty}^\infty \Big( \prod_{n = 1}^N p(x_n) \Big) \: \delta\Big( S - \sum_{n = 1}^N x_n \Big) \dd{x_1} \cdots \dd{x_N} + p(t) + &= \int\cdots\int_{-\infty}^\infty \Big( \prod_{n = 1}^N p(x_n) \Big) \: \delta\Big( t - \sum_{n = 1}^N x_n \Big) \dd{x_1} \cdots \dd{x_N} \\ - &= \Big( p_1 * \big( p_2 * ( ... * (p_N * \delta))\big)\Big)(S) + &= \Big( p_1 * \big( p_2 * ( ... * (p_N * \delta))\big)\Big)(t) \end{aligned}$$ In other words, the integrals pick out all combinations of $$x_n$$ which -add up to the desired $$S$$-value, and multiply the probabilities +add up to the desired $$t$$-value, and multiply the probabilities $$p(x_1) p(x_2) \cdots p(x_N)$$ of each such case. This is a convolution, so the [convolution theorem](/know/concept/convolution-theorem/) states that it is a product in the Fourier domain: $$\begin{aligned} - \phi_S(k) = \prod_{n = 1}^N \phi_n(k) + \phi_t(k) + = \prod_{n = 1}^N \phi_n(k) \end{aligned}$$ By taking the logarithm of both sides, the product becomes a sum, which we further expand: $$\begin{aligned} - \ln\!\big(\phi_S(k)\big) + \ln\!\big(\phi_t(k)\big) = \sum_{n = 1}^N \ln\!\big(\phi_n(k)\big) = \sum_{n = 1}^N \sum_{m = 1}^{\infty} \frac{(ik)^m}{m!} C_n^{(m)} \end{aligned}$$ -Consequently, the cumulants $$C^{(m)}$$ stack additively for the sum $$S$$ +Consequently, the cumulants $$C^{(m)}$$ stack additively for the sum $$t$$ of independent variables $$x_m$$, and therefore the means $$C^{(1)}$$ and variances $$C^{(2)}$$ do too: $$\begin{aligned} - C_S^{(m)} = \sum_{n = 1}^N C_n^{(m)} = C_1^{(m)} + C_2^{(m)} + ... + C_N^{(m)} + C_t^{(m)} + = \sum_{n = 1}^N C_n^{(m)} + = C_1^{(m)} + C_2^{(m)} + ... + C_N^{(m)} \end{aligned}$$ We now introduce the scaled sum $$z$$ as the new combined variable: $$\begin{aligned} - z = \frac{S}{\sqrt{N}} = \frac{1}{\sqrt{N}} (x_1 + x_2 + ... + x_N) + z + \equiv \frac{t}{\sqrt{N}} + = \frac{1}{\sqrt{N}} (x_1 + x_2 + ... + x_N) \end{aligned}$$ Its characteristic function $$\phi_z(k)$$ is then as follows, @@ -176,28 +188,30 @@ For sufficiently large $$N$$, we can therefore approximate it using just the fir $$\begin{aligned} \ln\!\big( \phi_z(k) \big) &\approx i k C^{(1)} - \frac{k^2}{2} C^{(2)} - = i k \overline{z} - \frac{k^2}{2} \sigma_z^2 + = i k \mu_z - \frac{k^2}{2} \sigma_z^2 \\ + \implies \quad \phi_z(k) - &\approx \exp(i k \overline{z}) \exp(- k^2 \sigma_z^2 / 2) + &\approx \exp(i k \mu_z) \exp(- k^2 \sigma_z^2 / 2) \end{aligned}$$ We take its inverse Fourier transform to get the density $$p(z)$$, -which turns out to be a Gaussian normal distribution, -which is even already normalized: +which turns out to be a Gaussian normal distribution +and is even already normalized: $$\begin{aligned} p(z) = \hat{\mathcal{F}}^{-1} \{\phi_z(k)\} - &= \frac{1}{2 \pi} \int_{-\infty}^\infty \exp\!\big(\!-\! i k (z - \overline{z})\big) \exp(- k^2 \sigma_z^2 / 2) \dd{k} + &= \frac{1}{2 \pi} \int_{-\infty}^\infty \exp\!\big(\!-\! i k (z - \mu_z)\big) \exp(- k^2 \sigma_z^2 / 2) \dd{k} \\ - &= \frac{1}{\sqrt{2 \pi \sigma_z^2}} \exp\!\Big(\!-\! \frac{(z - \overline{z})^2}{2 \sigma_z^2} \Big) + &= \frac{1}{\sqrt{2 \pi \sigma_z^2}} \exp\!\Big(\!-\! \frac{(z - \mu_z)^2}{2 \sigma_z^2} \Big) \end{aligned}$$ Therefore, the sum of many independent variables tends to a normal distribution, regardless of the densities of the individual variables. + ## References 1. H. Gould, J. Tobochnik, *Statistical and thermal physics*, 2nd edition, -- cgit v1.2.3