improve readability and equation rendering

Author: HugoMVale

doc/specs/stdlib_stats_distribution_normal.md

@@ -14,13 +14,13 @@ Experimental
 
 ### Description
 
-A normal continuous random variate distribution, also known as Gaussian, or Gauss or Laplace-Gauss distribution. The location `loc` specifies the mean or expectation. The `scale` specifies the standard deviation. 
+A normal continuous random variate distribution, also known as Gaussian, or Gauss or Laplace-Gauss distribution. The location `loc` specifies the mean or expectation ($\mu$). The `scale` specifies the standard deviation ($\sigma$). 
 
-Without argument the function returns a standard normal distributed random variate N(0,1).
+Without argument, the function returns a standard normal distributed random variate $N(0,1)$.
 
-With two arguments, the function returns a normal distributed random variate N(loc, scale^2). For complex arguments, the real and imaginary parts are independent of each other.
+With two arguments, the function returns a normal distributed random variate $N(\mu=\text{loc}, \sigma^2=\text{scale}^2)$. For complex arguments, the real and imaginary parts are independent of each other.
 
-With three arguments, the function returns a rank one array of normal distributed random variates.
+With three arguments, the function returns a rank-1 array of normal distributed random variates.
 
 Note: the algorithm used for generating normal random variates is fundamentally limited to double precision.
 
@@ -44,7 +44,7 @@ Elemental function (passing both `loc` and `scale`).
 
 ### Return value
 
-The result is a scalar or rank one array, with a size of `array_size`, and as the same type of `scale` and `loc`.
+The result is a scalar or rank-1 array, with a size of `array_size`, and as the same type of `scale` and `loc`.
 
 ### Example
 
@@ -62,11 +62,11 @@ Experimental
 
 The probability density function (pdf) of the single real variable normal distribution:
 
-$$f(x) = \frac{1}{\sigma \sqrt{2}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}$$
+$$f(x) = \frac{1}{\sigma \sqrt{2}} \exp{\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right]}$$
 
-For complex varible (x + y i) with independent real x and imaginary y parts, the joint probability density function is the product of corresponding marginal pdf of real and imaginary pdf (ref. "Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):
+For a complex varible $z=(x + y i)$ with independent real $x$ and imaginary $y$ parts, the joint probability density function is the product of the the corresponding real and imaginary marginal pdfs (ref. "Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):
 
-$$f(x + y \mathit{i}) = f(x) f(y) = \frac{1}{2\sigma_{x}\sigma_{y}} e^{-\frac{1}{2}[(\frac{x-\mu}{\sigma_{x}})^{2}+(\frac{y-\nu}{\sigma_{y}})^{2}]}$$
+$$f(x + y \mathit{i}) = f(x) f(y) = \frac{1}{2\sigma_{x}\sigma_{y}} \exp{\left[-\frac{1}{2}\left(\left(\frac{x-\mu_x}{\sigma_{x}}\right)^{2}+\left(\frac{y-\mu_y}{\sigma_{y}}\right)^{2}\right)\right]}$$
 
 ### Syntax
 
@@ -88,7 +88,7 @@ All three arguments must have the same type.
 
 ### Return value
 
-The result is a scalar or an array, with a shape conformable to arguments, and as the same type of input arguments.
+The result is a scalar or an array, with a shape conformable to the arguments, and of the same type as the input arguments.
 
 ### Example
 
@@ -106,11 +106,13 @@ Experimental
 
 Cumulative distribution function of the single real variable normal distribution:
 
-$$F(x)=\frac{1}{2}\left [ 1+erf(\frac{x-\mu}{\sigma \sqrt{2}}) \right ]$$
+$$F(x) = \frac{1}{2}\left [ 1+\text{erf}\left(\frac{x-\mu}{\sigma \sqrt{2}}\right) \right ]$$
 
-For the complex variable (x + y i) with independent real x and imaginary y parts, the joint cumulative distribution function is the product of corresponding marginal cdf of real and imaginary cdf (ref. "Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):
+For the complex variable $z=(x + y i)$ with independent real $x$ and imaginary $y$ parts, the joint cumulative distribution function is the product of the corresponding real and imaginary marginal cdfs (ref. "Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):
 
-$$F(x+y\mathit{i})=F(x)F(y)=\frac{1}{4} [1+erf(\frac{x-\mu}{\sigma_{x} \sqrt{2}})] [1+erf(\frac{y-\nu}{\sigma_{y} \sqrt{2}})]$$
+$$ F(x+y\mathit{i})=F(x)F(y)=\frac{1}{4} \
+\left[ 1+\text{erf}\left(\frac{x-\mu_x}{\sigma_x \sqrt{2}}\right) \right] \
+\left[ 1+\text{erf}\left(\frac{y-\mu_y}{\sigma_y \sqrt{2}}\right) \right] $$
 
 ### Syntax