improve readibility, fix equation rendering, update specs

Author: HugoMVale

doc/specs/stdlib_stats_distribution_exponential.md

@@ -14,34 +14,37 @@ Experimental
 
 ### Description
 
-An exponential distribution is the distribution of time between events in a Poisson point process. The inverse scale parameter `lambda` specifies the average time between events, also called the rate of events.
+An exponential distribution is the distribution of time between events in a Poisson point process. The inverse scale parameter `lambda` specifies the average time between events ($\lambda$), also called the rate of events.
 
-Without argument the function returns a random sample from the standard exponential distribution `E(1)` with `lambda = 1`.
+Without argument, the function returns a random sample from the standard exponential distribution $E(\lambda=1)$.
 
-With a single argument, the function returns a random sample from the exponential distribution `E(lambda)`.
+With a single argument, the function returns a random sample from the exponential distribution $E(\lambda=\text{lambda})$.
 For complex arguments, the real and imaginary parts are sampled independently of each other.
 
-With two arguments the function returns a rank one array of exponentially distributed random variates.
+With two arguments, the function returns a rank-1 array of exponentially distributed random variates.
 
-Note: the algorithm used for generating exponetial random variates is fundamentally limited to double precision. Ref.: Marsaglia, G. & Tsang, W.W. (2000) `The ziggurat method for generating random variables', J. Statist. Software, v5(8).
+@note
+The algorithm used for generating exponential random variates is fundamentally limited to double precision.[^1]
 
 ### Syntax
 
 `result = [[stdlib_stats_distribution_exponential(module):rvs_exp(interface)]]([lambda] [[, array_size]])`
 
 ### Class
 
-Function
+Elemental function
 
 ### Arguments
 
-`lambda`: optional argument has `intent(in)` and is a scalar of type `real` or `complex`. The value of `lambda` has to be non-negative.
+`lambda`: optional argument has `intent(in)` and is a scalar of type `real` or `complex`.
+If `lambda` is `real`, its value must be positive. If `lambda` is `complex`, both the real and imaginary components must be positive.
 
 `array_size`: optional argument has `intent(in)` and is a scalar of type `integer` with default kind.
 
 ### Return value
 
-The result is a scalar or rank one array with a size of `array_size`, and has the same type of `lambda`.
+The result is a scalar or rank-1 array with a size of `array_size`, and the same type as `lambda`.
+If `lambda` is non-positive, the result is `NaN`.
 
 ### Example
 
@@ -57,15 +60,13 @@ Experimental
 
 ### Description
 
-The probability density function (pdf) of the single real variable exponential distribution:
+The probability density function (pdf) of the single real variable exponential distribution is:
 
-$$f(x)=\begin{cases} \lambda e^{-\lambda x} &x\geqslant 0 \\\\ 0 &x< 0\end{}$$
+$$f(x)=\begin{cases} \lambda e^{-\lambda x} &x\geqslant 0 \\\\ 0 &x< 0\end{cases}$$
 
-For a complex variable (x + y i) with independent real x and imaginary y parts, the joint probability density function
-is the product of the corresponding marginal pdf of real and imaginary pdf (for more details, see
-"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 variable $z=(x + y i)$ with independent real $x$ and imaginary $y$ parts, the joint probability density function is the product of the corresponding real and imaginary marginal pdfs:[^2]
 
-$$f(x+\mathit{i}y)=f(x)f(y)=\begin{cases} \lambda_{x} \lambda_{y} e^{-(\lambda_{x} x + \lambda_{y} y)} &x\geqslant 0, y\geqslant 0 \\\\ 0 &otherwise\end{}$$
+$$f(x+\mathit{i}y)=f(x)f(y)=\begin{cases} \lambda_{x} \lambda_{y} e^{-(\lambda_{x} x + \lambda_{y} y)} &x\geqslant 0, y\geqslant 0 \\\\ 0 &\text{otherwise}\end{cases}$$
 
 ### Syntax
 
@@ -80,12 +81,13 @@ Elemental function
 `x`: has `intent(in)` and is a scalar of type `real` or `complex`.
 
 `lambda`: has `intent(in)` and is a scalar of type `real` or `complex`.
+If `lambda` is `real`, its value must be positive. If `lambda` is `complex`, both the real and imaginary components must be positive.
 
 All arguments must have the same type.
 
 ### Return value
 
-The result is a scalar or an array, with a shape conformable to arguments, and has the same type of input arguments.
+The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. If `lambda` is non-positive, the result is `NaN`.
 
 ### Example
 
@@ -103,13 +105,11 @@ Experimental
 
 Cumulative distribution function (cdf) of the single real variable exponential distribution:
 
-$$F(x)=\begin{cases}1 - e^{-\lambda x} &x\geqslant 0 \\\\ 0 &x< 0\end{}$$
+$$F(x)=\begin{cases}1 - e^{-\lambda x} &x\geqslant 0 \\\\ 0 &x< 0\end{cases}$$
 
-For a 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 (for more details, see
-"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 variable $z=(x + y i)$ with independent real $x$ and imaginary $y$ parts, the joint cumulative distribution function is the product of corresponding real and imaginary marginal cdfs:[^2]
 
-$$F(x+\mathit{i}y)=F(x)F(y)=\begin{cases} (1 - e^{-\lambda_{x} x})(1 - e^{-\lambda_{y} y}) &x\geqslant 0, \;\; y\geqslant 0 \\\\ 0 &otherwise \end{}$$
+$$F(x+\mathit{i}y)=F(x)F(y)=\begin{cases} (1 - e^{-\lambda_{x} x})(1 - e^{-\lambda_{y} y}) &x\geqslant 0, \;\; y\geqslant 0 \\\\ 0 & \text{otherwise} \end{cases}$$
 
 ### Syntax
 
@@ -124,15 +124,20 @@ Elemental function
 `x`: has `intent(in)` and is a scalar of type `real` or `complex`.
 
 `lambda`: has `intent(in)` and is a scalar of type `real` or `complex`.
+If `lambda` is `real`, its value must be positive. If `lambda` is `complex`, both the real and imaginary components must be positive.
 
 All arguments must have the same type.
 
 ### Return value
 
-The result is a scalar or an array, with a shape conformable to arguments, and has the same type of input arguments.
+The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. If `lambda` is non-positive, the result is `NaN`.
 
 ### Example
 
 ```fortran
 {!example/stats_distribution_exponential/example_exponential_cdf.f90!}
 ```
+
+[^1] Marsaglia, George, and Wai Wan Tsang. "The ziggurat method for generating random variables." _Journal of statistical software_ 5 (2000): 1-7.
+
+[^2] Miller, Scott, and Donald Childers. _Probability and random processes: With applications to signal processing and communications_. Academic Press, 2012 (p. 197).