Random numbers

The goal of this task is to generate a collection filled with 1000 normally distributed random (or pseudorandom) numbers with a mean of 1.0 and a standard deviation of 0.5

Many libraries only generate uniformly distributed random numbers. If so, use this formula to convert them to a normal distribution.

ALGOL 68

Translation of: C

Works with: ALGOL 68 version Revision 1 - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

<lang algol68>PROC random normal = REAL: # normal distribution, centered on 0, std dev 1 # (

 sqrt(-2*log(random)) * cos(2*pi*random)

);

test:(

 [1000]REAL rands;
 FOR i TO UPB rands DO
   rands[i] := 1 + random normal/2
 OD;
 INT limit=10;
 printf(($"("n(limit-1)(-d.6d",")-d.5d" ... )"$, rands[:limit]))

)</lang> Output sample:

( 0.693461, 0.948424, 0.482261, 1.045939, 0.890818, 1.467935, 0.604153, 0.804811, 0.690227, 0.83462 ... )

AWK

<lang awk>$ awk 'func r(){return sqrt(-2*log(rand()))*cos(6.2831853*rand())}BEGIN{for(i=0;i<1000;i++)s=s" "1+0.5*r();print s}'</lang>

BASIC

Works with: QuickBasic version 4.5

RANDOMIZE TIMER 'seeds random number generator with the system time
pi = 3.141592653589793#
DIM a(1 TO 1000) AS DOUBLE
CLS
FOR i = 1 TO 1000
   a(i) = 1 + SQR(-2 * LOG(RND)) * COS(2 * pi * RND)
NEXT i

This is the Javascript translation into C#. <lang csharp> private static double randomNormal() { return Math.Cos(2 * Math.PI * tRand.NextDouble()) * Math.Sqrt(-2 * Math.Log(tRand.NextDouble())); } </lang>

Then the methods in Metafont are used to calculate the average and the Standard Derivation: <lang csharp> static Random tRand = new Random();

static void Main(string[] args) { double[] a = new double[1000];

double tAvg = 0; for (int x = 0; x < a.Length; x++) { a[x] = randomNormal() / 2 + 1; tAvg += a[x]; }

tAvg /= a.Length; Console.WriteLine("Average: " + tAvg.ToString());

double s = 0; for (int x = 0; x < a.Length; x++) { s += Math.Pow((a[x] - tAvg), 2); } s = Math.Sqrt(s / 1000);

Console.WriteLine("Standard Derivative: " + s.ToString());

Console.ReadLine(); } </lang>

An example result:

Average: 1,00510073053613
Standard Derivative: 0,502540443430955

C++

<lang cpp>#include <cstdlib> // for rand

include <cmath> // for atan, sqrt, log, cos
include <algorithm> // for generate_n

double const pi = 4*std::atan(1.0);

// simple functor for normal distribution class normal_distribution { public:

 normal_distribution(double m, double s): mu(m), sigma(s) {}
 double operator() // returns a single normally distributed number
 {
   double r1 = (std::rand() + 1.0)/(RAND_MAX + 1.0); // gives equal distribution in (0, 1]
   double r2 = (std::rand() + 1.0)/(RAND_MAX + 1.0);
   return mu + sigma * std::sqrt(-2*std::log(r1))*std::cos(2*pi*r2);
 }

private:

 double mu, sigma;

};

int main() {

 double array[1000];
 std::generate_n(array, 1000, normal_distribution(1.0, 0.5));
 return 0;

}</lang>

Library: Boost

This example used Mersenne Twister generator. It can be changed by changing the typedef.

include <vector>
include "boost/random.hpp"
include "boost/generator_iterator.hpp"
include <boost/random/normal_distribution.hpp>
include <algorithm>

typedef boost::mt19937 RNGType; ///< mersenne twister generator

int main() {

   RNGType rng;
   boost::normal_distribution<> rdist(1.0,0.5); /**< normal distribution 
                          with mean of 1.0 and standard deviation of 0.5 */

   boost::variate_generator< RNGType, boost::normal_distribution<> >
                   get_rand(rng, rdist);

   std::vector<double> v(1000);
   generate(v.begin(),v.end(),get_rand);
   return 0;

} </lang>

Clojure

<lang lisp>(import '(java.util Random)) (def normals

 (let [r (Random.)]
   (take 1000 (repeatedly #(-> r .nextGaussian (* 0.5) (+ 1.0))))))</lang>

Common Lisp

<lang lisp>(loop for i from 1 to 1000

     collect (1+ (* (sqrt (* -2 (log (random 1.0)))) (cos (* 2 pi (random 1.0))) 0.5)))</lang>

D

Library: tango

<lang d>import tango.math.random.Random; double [1000]list; auto r = new Random(); foreach(ref l; list) {

 r.normalSource!(double)()(l);
 l = 1.0 + 0.5 * l;

}</lang>

E

<lang e>accum [] for _ in 1..1000 { _.with(entropy.nextGaussian()) }</lang>

Factor

<lang factor>1000 [ 1.0 0.5 normal-random-float ] replicate</lang>

Falcon

<lang falcon>a = [] for i in [0:1000] : a+= norm_rand_num()

function norm_rand_num()

  pi = 2*acos(0)
  return 1 + (cos(2 * pi * random()) * pow(-2 * log(random()) ,1/2)) /2

end</lang>

Forth

Works with: gforth version 0.6.2

<lang forth>require random.fs here to seed

-1. 1 rshift 2constant MAX-D \ or s" MAX-D" ENVIRONMENT? drop

frnd ( -- f ) \ uniform distribution 0..1

 rnd rnd dabs d>f MAX-D d>f f/ ;

frnd-normal ( -- f ) \ centered on 0, std dev 1

 frnd pi f* 2e f* fcos
 frnd fln -2e f* fsqrt f* ;

,normals ( n -- ) \ store many, centered on 1, std dev 0.5

 0 do frnd-normal 0.5e f* 1e f+ f, loop ;

create rnd-array 1000 ,normals</lang>

Fortran

Works with: Fortran version 90 and later

<lang fortran>PROGRAM Random

 INTEGER, PARAMETER :: n = 1000
 INTEGER :: i
 REAL :: array(n), pi, temp, mean = 1.0, sd = 0.5

 pi = 4.0*ATAN(1.0)
 CALL RANDOM_NUMBER(array) ! Uniform distribution

! Now convert to normal distribution

 DO i = 1, n-1, 2
   temp = sd * SQRT(-2.0*LOG(array(i))) * COS(2*pi*array(i+1)) + mean
   array(i+1) = sd * SQRT(-2.0*LOG(array(i))) * SIN(2*pi*array(i+1)) + mean
   array(i) = temp
 END DO

! Check mean and standard deviation

 mean = SUM(array)/n
 sd = SQRT(SUM((array - mean)**2)/n)

 WRITE(*, "(A,F8.6)") "Mean = ", mean
 WRITE(*, "(A,F8.6)") "Standard Deviation = ", sd

END PROGRAM Random</lang>

Output

Mean = 0.995112
Standard Deviation = 0.503373

F#

<lang fsharp>let gaussianRand count =

   let o = new System.Random()
   let pi = System.Math.PI
   let gaussrnd = 
       (fun _ -> 1. + 0.5 * sqrt(-2. * log(o.NextDouble())) * cos(2. * pi * o.NextDouble()))
   [ for i in {0 .. (int count)} -> gaussrnd() ]</lang>

Go

<lang go>package main import "rand" import "time"

func main() {

 var list [1000]float64
 rand.Seed(time.Nanoseconds())
 for i := range list {
   list[i] = 1.0 + 0.5 * rand.NormFloat64()
 }

}</lang>

Groovy

<lang groovy>rnd = new Random() result = (1..1000).inject([]) { r, i -> r << rnd.nextGaussian() }</lang>

Haskell

<lang haskell>import System.Random

pairs :: [a] -> [(a,a)] pairs (x:y:zs) = (x,y):pairs zs pairs _ = []

gauss mu sigma (r1,r2) =

 mu + sigma * sqrt (-2 * log r1) * cos (2 * pi * r2)

gaussians :: (RandomGen g, Random a, Floating a) => Int -> g -> [a] gaussians n g = take n $ map (gauss 1.0 0.5) $ pairs $ randoms g

result :: IO [Double] result = getStdGen >>= \g -> return $ gaussians 1000 g</lang>

HicEst

<lang hicest>REAL :: n=1000, m=1, s=0.5, array(n)

pi = 4 * ATAN(1) array = s * (-2*LOG(RAN(1)))^0.5 * COS(2*pi*RAN(1)) + m </lang>

Icon and Unicon

The seed &random may be assigned in either language; either to randomly seed or to pick a fixed starting point. ?i is the random number generator, returning an integer from 0 to i - 1 for non-zero integer i. As a special case, ?0 yields a random floating point number from 0.0 <= r < 1.0

Icon

<lang icon> procedure main()

   local L
   L := list(1000)
   every L[1 to 1000] := 1.0 + 0.5 * sqrt(-2.0 * log(?0)) * cos(2.0 * &pi * ?0)

   every write(!L)

end </lang>

Unicon

The solution is identical in Icon and Unicon, but note that Unicon randomly seeds it's generator.

IDL

<lang idl>result = 1.0 + 0.5*randomn(seed,1000)</lang>

J

Solution: <lang j>urand=: ?@$ 0: zrand=: (2 o. 2p1 * urand) * [: %: _2 * [: ^. urand

1 + 0.5 * zrand 100</lang>

Alternative Solution:
Using the normal script from the stats/distribs addon. <lang j> require 'stats/distribs/normal'

  1 0.5 rnorm 1000

1.44868803 1.21548637 0.812460657 1.54295452 1.2470606 ...</lang>

Java

<lang java>double[] list = new double[1000]; Random rng = new Random(); for(int i = 0;i<list.length;i++) {

 list[i] = 1.0 + 0.5 * rng.nextGaussian()

}</lang>

JavaScript

<lang javascript>function randomNormal() {

 return Math.cos(2 * Math.PI * Math.random()) * Math.sqrt(-2 * Math.log(Math.random()));

}

var a = new Array(1000); for (var i=0; i<a.length; i++)

 a[i] = randomNormal() / 2 + 1;</lang>

Liberty BASIC

<lang lb>dim a(1000) mean =1 sd =0.5 for i = 1 to 1000 ' throw 1000 normal variates

  a( i)  =mean +sd *( sqr( -2 * log( rnd( 0))) * cos( 2 * pi * rnd( 0)))

next i</lang>

Logo

Works with: UCB Logo

The earliest Logos only have a RANDOM function for picking a random non-negative integer. Many modern Logos have floating point random generators built-in. <lang logo>to random.float ; 0..1

 localmake "max.int lshift -1 -1
 output quotient random :max.int :max.int

end

to random.gaussian

 output product cos random 360  sqrt -2 / ln random.float

end

make "randoms cascade 1000 [fput random.gaussian / 2 + 1 ?] []</lang>

lua

<lang lua> local list = {} for i = 1, 1000 do

 list[i] = 1 + math.sqrt(-2 * math.log(math.random())) * math.cos(2 * math.pi * math.random()) / 2

end </lang>

Mathematica

Built-in function RandomReal with built-in distribution NormalDistribution as an argument: <lang Mathematica>RandomReal[NormalDistribution[1, 1/2], 1000]</lang>

MATLAB

This script uses the Box-Mueller Transform to transform a number from the uniform distribution to a normal distribution of mean = mu0 and standard deviation = chi2.

<lang MATLAB>function randNum = randNorm(mu0,chi2)

   radiusSquared = +Inf;

   while (radiusSquared >= 1)
       u = ( 2 * rand(1) ) - 1;
       v = ( 2 * rand(1) ) - 1;

       radiusSquared = u^2 + v^2;
   end

   scaleFactor = sqrt( ( -2*log(radiusSquared) )/ radiusSquared );
   randNum = (v * scaleFactor * chi2) + mu0;

end</lang>

Output: <lang MATLAB>>> randNorm(1,.5)

ans =

  0.693984121077029</lang>

MAXScript

<lang maxscript>arr = #() for i in 1 to 1000 do (

   a = random 0.0 1.0
   b = random 0.0 1.0
   c = 1.0 + 0.5 * sqrt (-2*log a) * cos (360*b) -- Maxscript cos takes degrees
   append arr c

)</lang>

Metafont

Metafont has normaldeviate which produces pseudorandom normal distributed numbers with mean 0 and variance one. So the following complete the task:

<lang metafont>numeric col[];

m := 0; % m holds the mean, for testing purposes for i = 1 upto 1000:

 col[i] := 1 + .5normaldeviate;
 m := m + col[i];

endfor

% testing m := m / 1000; % finalize the computation of the mean

s := 0; % in s we compute the standard deviation for i = 1 upto 1000:

 s := s + (col[i] - m)**2;

endfor s := sqrt(s / 1000);

show m, s; % and let's show that really they get what we wanted end</lang>

A run gave

>> 0.99947
>> 0.50533

Assigning a value to the special variable randomseed will allow to have always the same sequence of pseudorandom numbers

Modula-3

Translation of: C

<lang modula3>MODULE Rand EXPORTS Main;

IMPORT Random; FROM Math IMPORT log, cos, sqrt, Pi;

VAR rands: ARRAY [1..1000] OF LONGREAL;

(* Normal distribution. *) PROCEDURE RandNorm(): LONGREAL =

 BEGIN
   WITH rand = NEW(Random.Default).init() DO
     RETURN 
       sqrt(-2.0D0 * log(rand.longreal())) * cos(2.0D0 * Pi * rand.longreal());
   END;
 END RandNorm;

BEGIN

 FOR i := FIRST(rands) TO LAST(rands) DO
   rands[i] := 1.0D0 + 0.5D0 * RandNorm();
 END;

END Rand.</lang>

Objeck

<lang objeck> bundle Default {

 class RandomNumbers {
   function : Main(args : String[]) ~ Nil {
     rands := Float->New[1000];
     for(i := 0; i < rands->Size(); i += 1;) {
       rands[i] := 1.0 + 0.5 * RandomNormal();
     };

     each(i : rands) {
       rands[i]->PrintLine();
     };
   }

   function : native : RandomNormal() ~ Float {
     return (2 * Float->Pi() * Float->Random())->Cos() * (-2 * (Float->Random()->Log()))->SquareRoot();
   }
 }

} </lang>

OCaml

<lang ocaml>let pi = 4. *. atan 1.;; let random_gaussian () =

 1. +. sqrt (-2. *. log (Random.float 1.)) *. cos (2. *. pi *. Random.float 1.);;

let a = Array.init 1000 (fun _ -> random_gaussian ());;</lang>

Octave

<lang octave>p = normrnd(1.0, 0.5, 1000, 1); disp(mean(p)); disp(sqrt(sum((p - mean(p)).^2)/numel(p));</lang>

Output:

1.0209
0.51048

Perl

<lang perl>my $PI = 2 * atan2 1, 0;

my @nums = map {

   1 + 0.5 * sqrt(-2 * log rand) * cos(2 * $PI * rand)

} 1..1000;</lang>

Perl 6

Works with: Rakudo version #22 "Thousand Oaks"

<lang perl6>sub randnorm ($mean, $stddev) {

   $mean + $stddev * sqrt(-2 * log rand) * cos(2 * pi * rand)

}

my @nums = map { randnorm 1, 0.5 }, ^1000;</lang>

PHP

<lang php>function random() {

   return mt_rand() / mt_getrandmax();

}

$pi = pi(); // Set PI

$a = array(); for ($i = 0; $i < 1000; $i++) {

   $a[$i] = 1.0 + ((sqrt(-2 * log(random())) * cos(2 * $pi * random())) * 0.5);

}</lang>

PicoLisp

Translation of: C

<lang PicoLisp>(load "@lib/math.l")

(de randomNormal () # Normal distribution, centered on 0, std dev 1

  (*/
     (sqrt (* -2.0 (log (rand 0 1.0))))
     (cos (*/ 2.0 pi (rand 0 1.0) `(* 1.0 1.0)))
     1.0 ) )

(seed (time)) # Randomize

(let Result

  (make                                           # Build list
     (do 1000                                     # of 1000 elements
        (link (+ 1.0 (/ (randomNormal) 2))) ) )
  (for N (head 7 Result)                          # Print first 7 results
     (prin (format N *Scl) " ") ) )</lang>

Output:

1.500334 1.212931 1.095283 0.433122 0.459116 1.302446 0.402477

PL/I

<lang PL/I> /* CONVERTED FROM WIKI FORTRAN */ Normal_Random: procedure options (main);

  declare (array(1000), pi, temp,
           mean initial (1.0), sd initial (0.5)) float (18);
  declare (i, n) fixed binary;
  
  n = hbound(array, 1);
  pi = 4.0*ATAN(1.0);
  array = random(); /* Uniform distribution */
  /* Now convert to normal distribution */
  DO i = 1 to n-1 by 2;
     temp = sd * SQRT(-2.0*LOG(array(i))) * COS(2*pi*array(i+1)) + mean;
     array(i+1) = sd * SQRT(-2.0*LOG(array(i))) * SIN(2*pi*array(i+1)) + mean;
     array(i) = temp;
  END;
  /* Check mean and standard deviation */
  mean = SUM(array)/n;
  sd = SQRT(SUM((array - mean)**2)/n);
  put skip edit ( "Mean = ", mean ) (a, F(18,16) );
  put skip edit ( "Standard Deviation = ", sd) (a, F(18,16));

END Normal_Random; </lang>

Pop11

<lang pop11>;;; Choose radians as arguments to trigonometic functions true -> popradians;

procedure generating standard normal distribution

define random_normal() -> result; lvars r1 = random0(1.0), r2 = random0(1.0);

    cos(2*pi*r1)*sqrt(-2*log(r2)) -> result

enddefine;

lvars array, i;

Put numbers on the stack

for i from 1 to 1000 do 1.0+0.5*random_normal() endfor;

collect them into array

consvector(1000) -> array;</lang>

PureBasic

<lang PureBasic>Procedure.f RandomNormal()

  ; This procedure can return any real number.
  Protected.f x1, x2

  ; random numbers from the open interval ]0, 1[
  x1 = (Random(999998)+1) / 1000000       ; must be > 0 because of Log(x1)
  x2 = (Random(999998)+1) / 1000000

  ProcedureReturn Sqr(-2*Log(x1)) * Cos(2*#PI*x2)

EndProcedure

Define i, n=1000

Dim a.f(n-1) For i = 0 To n-1

  a(i) = 1 + 0.5 * RandomNormal()

Next</lang>

Python

Works with: Python version 2.5

<lang python>import random values = [random.gauss(1, .5) for i in range(1000)]

or [ random.normalvariate(1, 0.5) for i in range(1000)]</lang>

Note that the random module in the Python standard library supports a number of statistical distribution methods.

Quick check <lang python>>>> mean = sum(values)/1000 >>> sdeviation = (sum((i - mean)**2 for i in values)/1000)**0.5 >>> mean, sdeviation (1.0127861555468178, 0.5006682783828207) </lang>

R

<lang r>result <- rnorm(1000, mean=1, sd=0.5)</lang>

Ruby

<lang ruby>Array.new(1000) { 1 + Math.sqrt(-2 * Math.log(rand)) * Math.cos(2 * Math::PI * rand) }</lang>

Sather

<lang sather>class MAIN is

 main is
   a:ARRAY{FLTD} := #(1000);
   i:INT;

   RND::seed(2010);
   loop i := 1.upto!(1000) - 1;
     a[i] := 1.0d + 0.5d * RND::standard_normal;
   end;

   -- testing the distribution
   mean ::= a.reduce(bind(_.plus(_))) / a.size.fltd;
   #OUT + "mean " + mean + "\n";
   a.map(bind(_.minus(mean)));
   a.map(bind(_.pow(2.0d)));
   dev ::= (a.reduce(bind(_.plus(_))) / a.size.fltd).sqrt;
   #OUT + "dev  " + dev + "\n";
 end;

end;</lang>

Scala

<lang scala>List.fill(1000)(1.0 + 0.5 * scala.util.Random.nextGaussian)</lang>

Standard ML

Works with: SML/NJ

SML/NJ has two structures for random numbers:

1) Rand (a linear congruential generator). You create the generator by calling Rand.mkRandom with a seed (of word type). You can call the generator with () repeatedly to get a word in the range [Rand.randMin, Rand.randMax]. You can use the Rand.norm function to transform the output into a real from 0 to 1, or use the Rand.range (i,j) function to transform the output into an int of the given range. <lang sml>val seed = 0w42; val gen = Rand.mkRandom seed; fun random_gaussian () =

 1.0 + Math.sqrt (~2.0 * Math.ln (Rand.norm (gen ()))) * Math.cos (2.0 * Math.pi * Rand.norm (gen ()));

val a = List.tabulate (1000, fn _ => random_gaussian ());</lang>

2) Random (a subtract-with-borrow generator). You create the generator by calling Random.rand with a seed (of a pair of ints). You can use the Random.randInt function to generate a random int over its whole range; Random.randNat to generate a non-negative random int; Random.randReal to generate a real between 0 and 1; or Random.randRange (i,j) to generate an int in the given range. <lang sml>val seed = (47,42); val gen = Random.rand seed; fun random_gaussian () =

 1.0 + Math.sqrt (~2.0 * Math.ln (Random.randReal gen)) * Math.cos (2.0 * Math.pi * Random.randReal gen);

val a = List.tabulate (1000, fn _ => random_gaussian ());</lang>

Other implementations of Standard ML have their own random number generators. For example, Moscow ML has a Random structure that is different from the one from SML/NJ.

Tcl

<lang tcl>package require Tcl 8.5 variable ::pi [expr acos(0)] proc ::tcl::mathfunc::nrand {} {

   expr {sqrt(-2*log(rand())) * cos(2*$::pi*rand())}

}

set mean 1.0 set stddev 0.5 for {set i 0} {$i < 1000} {incr i} {

   lappend result [expr {$mean + $stddev*nrand()}]

}</lang>

TI-83 BASIC

Calculator symbol translations:

"STO" arrow: →

Square root sign: √

ClrList L₁
Radian
For(A,1,1000)
√(-2*ln(rand))*cos(2*π*A)→L₁(A)
End

Ursala

There are two ways of interpreting the task, either to simulate sampling a population described by the given statistics, or to construct a sample exhibiting the given statistics. Both are illustrated below. The functions parameterized by the mean and standard deviation take a sample size and return a sample of that size, represented as a list of floating point numbers. The Z library function simulates a draw from a standard normal distribution. Mean and standard deviation library functions are also used in this example. <lang Ursala>#import nat

import flo

pop_stats("mu","sigma") = plus/*"mu"+ times/*"sigma"+ Z*+ iota

sample_stats("mu","sigma") = plus^*D(minus/"mu"+ mean,~&)+ vid^*D(div\"sigma"+ stdev,~&)+ Z*+ iota

cast %eWL

test =

^(mean,stdev)* <

  pop_stats(1.,0.5) 1000,
  sample_stats(1.,0.5) 1000></lang>

The output shows the mean and standard deviation for both sample vectors, the latter being exact by construction.

<
   (1.004504e+00,4.915525e-01),
   (1.000000e+00,5.000000e-01)>