modify for complex case

Author: Jim-215-Fisher

doc/specs/stdlib_stats_distribution_normal.md

@@ -102,10 +102,14 @@ Experimental
 
 ### Description
 
-The probability density function of the continuous normal distribution.
+The probability density function (pdf) of the single real variable normal distribution:
 
 $$f(x) = \frac{1}{\sigma \sqrt{2}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}$$
 
+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):
+
+$$f(x + y \mathit{i}) = f(x) f(y) = \frac{1}{2\sigma_{x}\sigma_{y}} e^{-\frac{1}{2}[(\frac{x-\mu}{\sigma_{x}})^{2}+(\frac{y-\nu}{\sigma_{y}})^{2}]}$$
+
 ### Syntax
 
 `result = [[stdlib_stats_distribution_normal(module):pdf_normal(interface)]](x, loc, scale)`
@@ -185,10 +189,14 @@ Experimental
 
 ### Description
 
-Cumulative distribution function of the normal continuous distribution
+Cumulative distribution function of the single real variable normal distribution:
 
 $$F(x)=\frac{1}{2}\left [ 1+erf(\frac{x-\mu}{\sigma \sqrt{2}}) \right ]$$
 
+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):
+
+$$F(x+y\mathit{i})=F(x)F(y)=\frac{1}{4} [1+erf(\frac{x-\mu}{\sigma_{x} \sqrt{2}})] [1+erf(\frac{y-\nu}{\sigma_{y} \sqrt{2}})]$$
+
 ### Syntax
 
 `result = [[stdlib_stats_distribution_normal(module):cdf_normal(interface)]](x, loc, scale)`

src/stdlib_stats_distribution_normal.fypp

@@ -74,7 +74,8 @@ contains
 
     subroutine zigset
     ! Marsaglia & Tsang generator for random normals & random exponentials.
-    ! Translated from C by Alan Miller ([email protected])
+    ! Translated from C by Alan Miller ([email protected]), released as public
+    ! domain (https://jblevins.org/mirror/amiller/)
     !
     ! Marsaglia, G. & Tsang, W.W. (2000) `The ziggurat method for generating
     ! random variables', J. Statist. Software, v5(8).

src/tests/stats/test_distribution_normal.fypp

@@ -2,7 +2,7 @@
 #:include "common.fypp"
 #:set RC_KINDS_TYPES = REAL_KINDS_TYPES + CMPLX_KINDS_TYPES
 program test_distribution_normal
-    use stdlib_kinds
+    use stdlib_kinds, only : sp, dp, qp
     use stdlib_error, only : check
     use stdlib_random, only : random_seed
     use stdlib_stats_distribution_uniform, only : uni => rvs_uniform
@@ -63,7 +63,7 @@ contains
            chisq = chisq + (freq(i) - expct) ** 2 / expct
         end do
         write(*,*) "The critical values for chi-squared with 1000 d. of f. is" &
-                    //" 1143.92"
+            //" 1143.92"
         write(*,*) "Chi-squared for normal random generator is : ", chisq
         call check((chisq < 1143.9),                                           &
             msg = "normal randomness failed chi-squared test", warn = warn)