Improve knowledge base

1 files changed, 14 insertions, 12 deletions

diff --git a/source/know/concept/central-limit-theorem/index.md b/source/know/concept/central-limit-theorem/index.md
index 42bc05b..0ebad36 100644
--- a/source/know/concept/central-limit-theorem/index.md
+++ b/source/know/concept/central-limit-theorem/index.md
@@ -17,7 +17,8 @@ and calculating $$M$$ averages $$\mu_m$$ (which involves summing over $$N$$),
 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 $$p(x_n)$$,
+More formally, for $$N$$ independent variables $$x_n$$
+with probability distributions $$p(x_n)$$,
 we define the following totals of all variables, means and variances:
 
 $$\begin{aligned}
@@ -39,9 +40,9 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-We prove this below,
-but first we need to introduce some tools.
-Given a probability density $$p(x)$$, its [Fourier transform](/know/concept/fourier-transform/)
+We prove this below, but first we need to introduce some tools.
+Given a probability density $$p(x)$$,
+its [Fourier transform](/know/concept/fourier-transform/)
 is called the **characteristic function** $$\phi(k)$$:
 
 $$\begin{aligned}
@@ -70,7 +71,8 @@ $$\begin{aligned}
     = i^n \: \overline{x^n}
 \end{aligned}$$
 
-Next, the **cumulants** $$C^{(n)}$$ are defined from the Taylor expansion of $$\ln\!\big(\phi(k)\big)$$:
+Next, the **cumulants** $$C^{(n)}$$ are defined
+from the Taylor expansion of $$\ln\!\big(\phi(k)\big)$$:
 
 $$\begin{aligned}
     \ln\!\big( \phi(k) \big)
@@ -96,9 +98,8 @@ $$\begin{aligned}
     = - \overline{x}^2 + \overline{x^2} = \sigma^2
 \end{aligned}$$
 
-Now that we have introduced these tools,
-we define $$t$$ as the sum
-of $$N$$ independent variables $$x_n$$, in other words:
+Now that we have introduced these tools, we repeat our definition of $$t$$
+as the sum of $$N$$ independent variables $$x_n$$, in other words:
 
 $$\begin{aligned}
     t
@@ -116,10 +117,11 @@ $$\begin{aligned}
     &= \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 $$t$$-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/)
+In other words, we integrate over all possible combinations of $$x_n$$,
+and use the Dirac delta function to pick out the combinations
+where the $$x_n$$ 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}