More improvements to knowledge base

1 files changed, 52 insertions, 38 deletions

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,