Random numbers
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Generate a collection filled with 1000 normally distributed random (or pseudo-random) numbers with a mean of 1.0 and a standard deviation of 0.5
Many libraries only generate uniformly distributed random numbers. If so, you may use one of these algorithms.
- Related task
Ada
with Ada.Numerics; use Ada.Numerics;
with Ada.Numerics.Float_Random; use Ada.Numerics.Float_Random;
with Ada.Numerics.Elementary_Functions; use Ada.Numerics.Elementary_Functions;
procedure Normal_Random is
function Normal_Distribution
( Seed : Generator;
Mu : Float := 1.0;
Sigma : Float := 0.5
) return Float is
begin
return
Mu + (Sigma * Sqrt (-2.0 * Log (Random (Seed), 10.0)) * Cos (2.0 * Pi * Random (Seed)));
end Normal_Distribution;
Seed : Generator;
Distribution : array (1..1_000) of Float;
begin
Reset (Seed);
for I in Distribution'Range loop
Distribution (I) := Normal_Distribution (Seed);
end loop;
end Normal_Random;
ALGOL 68
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]))
)- Output:
( 0.693461, 0.948424, 0.482261, 1.045939, 0.890818, 1.467935, 0.604153, 0.804811, 0.690227, 0.83462 ... )
Arturo
rnd: function []-> (random 0 10000)//10000
rands: map 1..1000 'x [
1 + (sqrt neg 2 * ln rnd) * (cos 2 * pi * rnd)
]
print rands
- Output:
0.6219537961087694 1.279396486161406 1.619019280815647 2.119538294228789 -0.1598383851981044 2.67797211803156 0.9304703037226587 1.629254364659528 -0.4171704717398712 0.9082342931486092 -0.5929704390625219 2.117000897984871 -0.1981633787460266 0.01132471973856153 2.102359263212924 0.2408823232884222 2.046195035792376 0.6374831627030295 0.000839808324124558 1.117061838266626 0.7413355299469649 0.4485598815755762 2.999434800016997 2.560580541932842 1.703197984879731 2.889159248353575 -1.800157205708138 1.756810020187321 0.7136708180852145 0.5929151678321705 0.332519993787973 2.660212054362758 0.5835660585480075 0.8527946892567934 1.640573993747053 0.09471843345263908 1.051402997891346 1.116149156137905 -0.7400139019343499 1.782572831979232 2.531779039786426 0.5240268064639871 0.07099232630526586 -0.854892656700071 1.54381929430469 -0.4416899008614745 0.4274356035015117 0.7350027625573482 2.153583935076981 1.461215281535983 -1.041723064151266 2.338060763553139 -0.1492967916030414 0.3799517724040202 0.4577924541353815 0.673317567666373 -2.27731583876462 1.28480355806061 -0.6925811023772748 -0.2642224122781984 0.6590513830891744 2.55537133425143 1.67933335469247 0.8659013395355968 -1.211026941441126 0.9524579534222226 -0.1931750631835656 -0.5119479869237693 0.1814749003063878 3.03139579963414 ...
AutoHotkey
contributed by Laszlo on the ahk forum
Loop 40
R .= RandN(1,0.5) "`n" ; mean = 1.0, standard deviation = 0.5
MsgBox %R%
RandN(m,s) { ; Normally distributed random numbers of mean = m, std.dev = s by Box-Muller method
Static i, Y
If (i := !i) { ; every other call
Random U, 0, 1.0
Random V, 0, 6.2831853071795862
U := sqrt(-2*ln(U))*s
Y := m + U*sin(V)
Return m + U*cos(V)
}
Return Y
}
Avail
Method "U(_,_)" is
[
lower : number,
upper : number
|
divisor ::= ((1<<32)) ÷ (upper - lower)→double;
map a pRNG through [i : integer | (i ÷ divisor) + lower]
];
Method "a Marsaglia polar sampler" is
[
generator for
[
yield : [double]→⊤
|
source ::= U(-1, 1);
Repeat [
x ::= take 1 from source[1];
y ::= take 1 from source[1];
s ::= x^2 + y^2;
If 0 < s < 1 then
[
factor ::= ((-2 × ln s) ÷ s) ^ 0.5;
yield(x × factor);
yield(y × factor);
];
]
]
];
// the default distribution has mean 0 and std dev 1.0, so we scale the values
sampler ::= map a Marsaglia polar sampler through [d : double | d ÷ 2.0 + 1.0];
values ::= take 1000 from sampler;AWK
One-liner:
$ 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}'
Readable version:
function r() {
return sqrt( -2*log( rand() ) ) * cos(6.2831853*rand() )
}
BEGIN {
n=1000
for(i=0;i<n;i++) {
x = 1 + 0.5*r()
s = s" "x
}
print s
}
- Output:
first few values only
0.783753 1.16682 1.17989 1.14975 1.34784 0.29296 0.979227 1.04402 0.567835 1.58812 0.465559 1.27186 0.324533 0.725827 -0.0626549 0.632273 1.0145 1.3387 0.861667 1.04147 1.2576 1.02497 0.58453 0.9619 1.26902 0.851048 -0.126259 0.863256
...
BASIC
ANSI BASIC
100 REM Random numbers
110 RANDOMIZE
120 DEF RandomNormal = COS(2 * PI * RND) * SQR(-2 * LOG(RND))
130 DIM R(0 TO 999)
140 LET Sum = 0
150 FOR I = 0 TO 999
160 LET R(I) = 1 + RandomNormal / 2
170 LET Sum = Sum + R(I)
180 NEXT I
190 LET Mean = Sum / 1000
200 LET Sum = 0
210 FOR I = 0 TO 999
220 LET Sum = Sum + (R(I) - Mean) ^ 2
230 NEXT I
240 LET SD = SQR(Sum / 1000)
250 PRINT "Mean is "; Mean
260 PRINT "Standard Deviation is"; SD
270 PRINT
280 END
- Output:
Two runs.
Mean is 1.00216454061435 Standard Deviation is .504515904812839
Mean is .995781408878628 Standard Deviation is .499307289407576
Applesoft BASIC
The Commodore BASIC code works in Applesoft BASIC.
BASIC256
# Generates normally distributed random numbers with mean 0 and standard deviation 1
function randomNormal()
return cos(2.0 * pi * rand) * sqr(-2.0 * log(rand))
end function
dim r(1000)
sum = 0.0
# Generate 1000 normally distributed random numbers
# with mean 1 and standard deviation 0.5
# and calculate their sum
for i = 0 to 999
r[i] = 1.0 + randomNormal() / 2.0
sum += r[i]
next i
mean = sum / 1000.0
sum = 0.0
# Now calculate their standard deviation
for i = 0 to 999
sum += (r[i] - mean) ^ 2.0
next i
sd = sqr(sum/1000.0)
print "Mean is "; mean
print "Standard Deviation is "; sd
end- Output:
Mean is 1.002092 Standard Deviation is 0.4838570687
BBC BASIC
DIM array(999)
FOR number% = 0 TO 999
array(number%) = 1.0 + 0.5 * SQR(-2*LN(RND(1))) * COS(2*PI*RND(1))
NEXT
mean = SUM(array()) / (DIM(array(),1) + 1)
array() -= mean
stdev = MOD(array()) / SQR(DIM(array(),1) + 1)
PRINT "Mean = " ; mean
PRINT "Standard deviation = " ; stdev
- Output:
Mean = 1.01848064 Standard deviation = 0.503551814
Chipmunk Basic
10 ' Random numbers
20 randomize timer
30 dim r(999)
40 sum = 0
50 for i = 0 to 999
60 r(i) = 1+randomnormal()/2
70 sum = sum+r(i)
80 next
90 mean = sum/1000
100 sum = 0
110 for i = 0 to 999
120 sum = sum+(r(i)-mean)^2
130 next
140 sd = sqr(sum/1000)
150 print "Mean is ";mean
160 print "Standard Deviation is ";sd
170 print
180 end
500 sub randomnormal()
510 randomnormal = cos(2*pi*rnd(1))*sqr(-2*log(rnd(1)))
520 end sub
- Output:
Two runs.
Mean is 1.007087 Standard Deviation is 0.496848
Mean is 0.9781 Standard Deviation is 0.508147
Commodore BASIC
10 DIM AR(999): DIM DE(999)
20 FOR N = 0 TO 999
30 AR(N)= 0 + SQR(-1.3*LOG(RND(1))) * COS(1.2*PI*RND(1))
40 NEXT N
50 :
60 REM SUM
70 LET SU = 0
80 FOR N = 0 TO 999
90 LET SU = SU + AR(N)
100 NEXT N
110 :
120 REM MEAN
130 LET ME= 0
140 LET ME = SU/1000
150 :
160 REM DEVIATION
170 FOR N = 0 TO 999
180 T = AR(N)-ME: REM SUBTRACT MEAN FROM NUMBER
190 T = T * T: REM SQUARE THE RESULT
200 DE(N) = T : REM STORE IN DEVIATION ARRAY
210 NEXT N
220 LET DS=0: REM SUM OF DEVIATION ARRAY
230 FOR N = 0 TO 999
240 LET DS = DS + DE(N)
250 NEXT N
260 LET DM=0: REM MEAN OF DEVIATION ARRAY
270 LET DM = DS / 1000
280 LET DE = 0:
290 LET DE = SQR(DM)
300 :
310 PRINT "MEAN = "ME
320 PRINT "STANDARD DEVIATION ="DE
330 END
FreeBASIC
' FB 1.05.0 Win64
Const pi As Double = 3.141592653589793
Randomize
' Generates normally distributed random numbers with mean 0 and standard deviation 1
Function randomNormal() As Double
Return Cos(2.0 * pi * Rnd) * Sqr(-2.0 * Log(Rnd))
End Function
Dim r(0 To 999) As Double
Dim sum As Double = 0.0
' Generate 1000 normally distributed random numbers
' with mean 1 and standard deviation 0.5
' and calculate their sum
For i As Integer = 0 To 999
r(i) = 1.0 + randomNormal/2.0
sum += r(i)
Next
Dim mean As Double = sum / 1000.0
Dim sd As Double
sum = 0.0
' Now calculate their standard deviation
For i As Integer = 0 To 999
sum += (r(i) - mean) ^ 2.0
Next
sd = Sqr(sum/1000.0)
Print "Mean is "; mean
Print "Standard Deviation is"; sd
Print
Print "Press any key to quit"
Sleep
Sample result:
- Output:
Mean is 1.000763573902885 Standard Deviation is 0.500653063426955
GW-BASIC
The Commodore BASIC code works in GW-BASIC.
Liberty BASIC
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 iMinimal BASIC
10 REM Random numbers
20 LET P = 4*ATN(1)
30 RANDOMIZE
40 DEF FNN = COS(2*P*RND)*SQR(-2*LOG(RND))
50 DIM R(999)
60 LET S = 0
70 FOR I = 0 TO 999
80 LET R(I) = 1+FNN/2
90 LET S = S+R(I)
100 NEXT I
110 LET M = S/1000
120 LET S = 0
130 FOR I = 0 TO 999
140 LET S = S+(R(I)-M)^2
150 NEXT I
160 LET D = SQR(S/1000)
170 PRINT "Mean is "; M
180 PRINT "Standard Deviation is"; D
190 PRINT
200 END
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.q(n-1)
For i = 0 To n-1
a(i) = 1 + 0.5 * RandomNormal()
Next
QuickBASIC
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
Run BASIC
dim a(1000)
pi = 22/7
for i = 1 to 1000
a( i) = 1 + .5 * (sqr(-2 * log(rnd(0))) * cos(2 * pi * rnd(0)))
next iTI-83 BASIC
Built-in function: randNorm()
randNorm(1,.5)
Or by a program:
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
ZX Spectrum Basic
Here we have converted the QBasic code to suit the ZX Spectrum:
10 RANDOMIZE 0 : REM seeds random number generator based on uptime
20 DIM a(1000)
30 CLS
40 FOR i = 1 TO 1000
50 LET a(i) = 1 + SQR(-2 * LN(RND)) * COS(2 * PI * RND)
60 NEXT i
C
#include <stdlib.h>
#include <math.h>
#ifndef M_PI
#define M_PI 3.14159265358979323846
#endif
double drand() /* uniform distribution, (0..1] */
{
return (rand()+1.0)/(RAND_MAX+1.0);
}
double random_normal() /* normal distribution, centered on 0, std dev 1 */
{
return sqrt(-2*log(drand())) * cos(2*M_PI*drand());
}
int main()
{
int i;
double rands[1000];
for (i=0; i<1000; i++)
rands[i] = 1.0 + 0.5*random_normal();
return 0;
}
C#
private static double randomNormal()
{
return Math.Cos(2 * Math.PI * tRand.NextDouble()) * Math.Sqrt(-2 * Math.Log(tRand.NextDouble()));
}
Then the methods in Random numbers#Metafont are used to calculate the average and the Standard Deviation:
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 Deviation: " + s.ToString());
Console.ReadLine();
}
An example result:
Average: 1,00510073053613 Standard Deviation: 0,502540443430955
C++
The new C++ standard looks very similar to the Boost library example below.
#include <random>
#include <functional>
#include <vector>
#include <algorithm>
using namespace std;
int main()
{
random_device seed;
mt19937 engine(seed());
normal_distribution<double> dist(1.0, 0.5);
auto rnd = bind(dist, engine);
vector<double> v(1000);
generate(v.begin(), v.end(), rnd);
return 0;
}
#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() const // 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:
const double mu, sigma;
};
int main()
{
double array[1000];
std::generate_n(array, 1000, normal_distribution(1.0, 0.5));
return 0;
}
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;
}
Clojure
(import '(java.util Random))
(def normals
(let [r (Random.)]
(take 1000 (repeatedly #(-> r .nextGaussian (* 0.5) (+ 1.0))))))
COBOL
IDENTIFICATION DIVISION.
PROGRAM-ID. RANDOM.
AUTHOR. Bill Gunshannon
INSTALLATION. Home.
DATE-WRITTEN. 14 January 2022.
************************************************************
** Program Abstract:
** Able to get the Mean to be really close to 1.0 but
** couldn't get the Standard Deviation any closer than
** .3 to .4.
************************************************************
DATA DIVISION.
WORKING-STORAGE SECTION.
01 Sample-Size PIC 9(5) VALUE 1000.
01 Total PIC 9(10)V9(5) VALUE 0.0.
01 Arith-Mean PIC 999V999 VALUE 0.0.
01 Std-Dev PIC 999V999 VALUE 0.0.
01 Seed PIC 999V999.
01 TI PIC 9(8).
01 Idx PIC 99999 VALUE 0.
01 Intermediate PIC 9(10)V9(5) VALUE 0.0.
01 Rnd-Work.
05 Rnd-Tbl
OCCURS 1 TO 99999 TIMES DEPENDING ON Sample-Size.
10 Rnd PIC 9V9999999 VALUE 0.0.
PROCEDURE DIVISION.
Main-Program.
ACCEPT TI FROM TIME.
MOVE FUNCTION RANDOM(TI) TO Seed.
PERFORM WITH TEST AFTER VARYING Idx
FROM 1 BY 1
UNTIL Idx = Sample-Size
COMPUTE Intermediate =
(FUNCTION RANDOM() * 2.01)
MOVE Intermediate TO Rnd(Idx)
END-PERFORM.
PERFORM WITH TEST AFTER VARYING Idx
FROM 1 BY 1
UNTIL Idx = Sample-Size
COMPUTE Total = Total + Rnd(Idx)
END-PERFORM.
COMPUTE Arith-Mean = Total / Sample-Size.
DISPLAY "Mean: " Arith-Mean.
PERFORM WITH TEST AFTER VARYING Idx
FROM 1 BY 1
UNTIL Idx = Sample-Size
COMPUTE Intermediate =
Intermediate + (Rnd(Idx) - Arith-Mean) ** 2
END-PERFORM.
COMPUTE Std-Dev = Intermediate / Sample-Size.
DISPLAY "Std-Dev: " Std-Dev.
STOP RUN.
END PROGRAM RANDOM.
Common Lisp
(loop for i from 1 to 1000
collect (1+ (* (sqrt (* -2 (log (random 1.0)))) (cos (* 2 pi (random 1.0))) 0.5)))
Crystal
n, mean, sd, tau = 1000, 1, 0.5, (2 * Math::PI)
array = Array.new(n) { mean + sd * Math.sqrt(-2 * Math.log(rand)) * Math.cos(tau * rand) }
mean = array.sum / array.size
standev = Math.sqrt( array.sum{ |x| (x - mean) ** 2 } / array.size )
puts "mean = #{mean}, standard deviation = #{standev}"
- Output:
mean = 1.0093442539237896, standard deviation = 0.504694489463623
D
import std.stdio, std.random, std.math;
struct NormalRandom {
double mean, stdDev;
// Necessary because it also defines an opCall.
this(in double mean_, in double stdDev_) pure nothrow {
this.mean = mean_;
this.stdDev = stdDev_;
}
double opCall() const /*nothrow*/ {
immutable r1 = uniform01, r2 = uniform01; // Not nothrow.
return mean + stdDev * sqrt(-2 * r1.log) * cos(2 * PI * r2);
}
}
void main() {
double[1000] array;
auto nRnd = NormalRandom(1.0, 0.5);
foreach (ref x; array)
//x = nRnd;
x = nRnd();
}
Alternative Version
(Untested)
import tango.math.random.Random;
void main() {
double[1000] list;
auto r = new Random();
foreach (ref l; list) {
r.normalSource!(double)()(l);
l = 1.0 + 0.5 * l;
}
}
Delphi
Delphi has RandG function which generates random numbers with normal distribution using Marsaglia-Bray algorithm:
program Randoms;
{$APPTYPE CONSOLE}
uses
Math;
var
Values: array[0..999] of Double;
I: Integer;
begin
// Randomize; Commented to obtain reproducible results
for I:= Low(Values) to High(Values) do
Values[I]:= RandG(1.0, 0.5); // Mean = 1.0, StdDev = 0.5
Writeln('Mean = ', Mean(Values):6:4);
Writeln('Std Deviation = ', StdDev(Values):6:4);
Readln;
end.
- Output:
Mean = 1.0098 Std deviation = 0.5016
DuckDB
(September 2, 2024)
In this entry, only five of the randomly generated values are shown, together with the whole-sample mean and standard deviation statistics as computed by the corresponding built-in aggregation functions.
The rnv() function is defined without reference to random() because of a current limitation of DuckDB's SQL.
# The Box-Muller method
create or replace function rnv(mean, sd, u, v) as
(select ( sqrt(-2 * ln(u)) * cos(2 * pi() * v) * sd) + mean);
create or replace table t as
(with rows as (select random() as u, random() as v from unnest(range(0,1000)))
select rnv(0, 0.5, u, v) as rnv from rows);
from t limit 5;
select avg(rnv) as average,
stddev_samp(rnv) as 'sample stddev',
stddev_pop(rnv) as 'population stddev'
from t;
- Output:
┌──────────────────────┐ │ rnv │ │ double │ ├──────────────────────┤ │ 0.2736446640135949 │ │ -0.14633937032848926 │ │ -0.6140022436594963 │ │ -0.47193674117986534 │ │ -0.8758486969986952 │ └──────────────────────┘ ┌──────────────────────┬────────────────────┬───────────────────┐ │ average │ sample stddev │ population stddev │ │ double │ double │ double │ ├──────────────────────┼────────────────────┼───────────────────┤ │ 0.004822768170835433 │ 0.4955201034287957 │ 0.495272281406079 │ └──────────────────────┴────────────────────┴───────────────────┘
DWScript
var values : array [0..999] of Float;
var i : Integer;
for i := values.Low to values.High do
values[i] := RandG(1, 0.5);
E
accum [] for _ in 1..1000 { _.with(entropy.nextGaussian()) }EasyLang
numfmt 5 0
e = 2.7182818284590452354
for i = 1 to 1000
a[] &= 1 + 0.5 * sqrt (-2 * log10 randomf / log10 e) * cos (360 * randomf)
.
for v in a[]
avg += v / len a[]
.
print "Average: " & avg
for v in a[]
s += pow (v - avg) 2
.
s = sqrt (s / len a[])
print "Std deviation: " & s
Eiffel
class
APPLICATION
inherit
ARGUMENTS
create
make
feature {NONE} -- Initialization
l_time: TIME
l_seed: INTEGER
math:DOUBLE_MATH
rnd:RANDOM
Size:INTEGER
once
Result:= 1000
end
make
-- Run application.
local
ergebnis:ARRAY[DOUBLE]
tavg: DOUBLE
x: INTEGER
tmp: DOUBLE
text : STRING
do
-- initialize random generator
create l_time.make_now
l_seed := l_time.hour
l_seed := l_seed * 60 + l_time.minute
l_seed := l_seed * 60 + l_time.second
l_seed := l_seed * 1000 + l_time.milli_second
create rnd.set_seed (l_seed)
-- initialize random number container and math
create ergebnis.make_filled (0.0, 1, size)
tavg := 0;
create math
from
x := 1
until
x > ergebnis.count
loop
tmp := randomNormal / 2 + 1
tavg := tavg + tmp
ergebnis.enter (tmp , x)
x := x + 1
end
tavg := tavg / ergebnis.count
text := "Average: "
text.append_double (tavg)
text.append ("%N")
print(text)
tmp := 0
from
x:= 1
until
x > ergebnis.count
loop
tmp := tmp + (ergebnis.item (x) - tavg)^2
x := x + 1
end
tmp := math.sqrt (tmp / ergebnis.count)
text := "Standard Deviation: "
text.append_double (tmp)
text.append ("%N")
print(text)
end
randomNormal:DOUBLE
local
first: DOUBLE
second: DOUBLE
do
rnd.forth
first := rnd.double_item
rnd.forth
second := rnd.double_item
Result := math.cosine (2 * math.pi * first) * math.sqrt (-2 * math.log (second))
end
end
Example Result
Average: 1.0079398405028137 Standard Deviation: 0.49042787564453988
Elena
ELENA 6.x :
import extensions;
import extensions'math;
randomNormal()
{
^ cos(2 * Pi_value * randomGenerator.nextReal())
* sqrt(-2 * ln(randomGenerator.nextReal()))
}
public program()
{
real[] a := new real[](1000);
real tAvg := 0;
for (int x := 0; x < a.Length; x += 1)
{
a[x] := (randomNormal()) / 2 + 1;
tAvg += a[x]
};
tAvg /= a.Length;
console.printLine("Average: ", tAvg);
real s := 0;
for (int x := 0; x < a.Length; x += 1)
{
s += power(a[x] - tAvg, 2)
};
s := sqrt(s / 1000);
console.printLine("Standard Deviation: ", s);
console.readChar()
}- Output:
Average: 0.9842420481571 Standard Deviation: 0.5109070975558
Elixir
defmodule Random do
def normal(mean, sd) do
{a, b} = {:rand.uniform, :rand.uniform}
mean + sd * (:math.sqrt(-2 * :math.log(a)) * :math.cos(2 * :math.pi * b))
end
end
std_dev = fn (list) ->
mean = Enum.sum(list) / length(list)
sd = Enum.reduce(list, 0, fn x,acc -> acc + (x-mean)*(x-mean) end) / length(list)
|> :math.sqrt
IO.puts "Mean: #{mean},\tStdDev: #{sd}"
end
xs = for _ <- 1..1000, do: Random.normal(1.0, 0.5)
std_dev.(xs)
- Output:
Mean: 1.009079383094275, StdDev: 0.4991894476975088
used Erlang function :rand.normal
xs = for _ <- 1..1000, do: 1.0 + :rand.normal * 0.5
std_dev.(xs)
- Output:
Mean: 0.9955701150615597, StdDev: 0.5036412260426065
Erlang
mean(Values) ->
mean(tl(Values), hd(Values), 1).
mean([], Acc, Length) ->
Acc / Length;
mean(Values, Acc, Length) ->
mean(tl(Values), hd(Values)+Acc, Length+1).
variance(Values) ->
Mean = mean(Values),
variance(Values, Mean, 0) / length(Values).
variance([], _, Acc) ->
Acc;
variance(Values, Mean, Acc) ->
Diff = hd(Values) - Mean,
DiffSqr = Diff * Diff,
variance(tl(Values), Mean, Acc + DiffSqr).
stddev(Values) ->
math:sqrt(variance(Values)).
normal(Mean, StdDev) ->
U = random:uniform(),
V = random:uniform(),
Mean + StdDev * ( math:sqrt(-2 * math:log(U)) * math:cos(2 * math:pi() * V) ). % Erlang's math:log is the natural logarithm.
main(_) ->
X = [ normal(1.0, 0.5) || _ <- lists:seq(1, 1000) ],
io:format("mean = ~w\n", [mean(X)]),
io:format("stddev = ~w\n", [stddev(X)]).
- Output:
mean = 1.0118289913718608 stddev = 0.5021636849524854
ERRE
PROGRAM DISTRIBUTION
!
! for rosettacode.org
!
! formulas taken from TI-59 Master Library manual
CONST NUM_ITEM=1000
!VAR SUMX#,SUMX2#,R1#,R2#,Z#,I%
DIM A#[1000]
BEGIN
! seeds random number generator with system time
RANDOMIZE(TIMER)
PRINT(CHR$(12);) !CLS
SUMX#=0 SUMX2#=0
FOR I%=1 TO NUM_ITEM DO
R1#=RND(1) R2#=RND(1)
Z#=SQR(-2*LOG(R1#))*COS(2*π*R2#)
A#[I%]=Z#/2+1 ! I want a normal distribution with
! mean=1 and std.dev=0.5
SUMX#+=A#[I%] SUMX2#+=A#[I%]*A#[I%]
END FOR
Z#=SUMX#/NUM_ITEM
PRINT("Average is";Z#)
PRINT("Standard dev. is";SQR(SUMX2#/NUM_ITEM-Z#*Z#))
END PROGRAM
Euler Math Toolbox
>v=normal(1,1000)*0.5+1;
>mean(v), dev(v)
1.00291801071
0.498226876528Euphoria
include misc.e
function RandomNormal()
atom x1, x2
x1 = rand(999999) / 1000000
x2 = rand(999999) / 1000000
return sqrt(-2*log(x1)) * cos(2*PI*x2)
end function
constant n = 1000
sequence s
s = repeat(0,n)
for i = 1 to n do
s[i] = 1 + 0.5 * RandomNormal()
end forF#
let n = MathNet.Numerics.Distributions.Normal(1.0,0.5)
List.init 1000 (fun _->n.Sample())
- Output:
[0.734433576; 1.54225304; 0.4407528678; 1.177675412; 0.4318617021; 0.6026656337; 0.769764924; 1.104693934; 0.6297500925; 0.9594598077; 1.684736389; 1.160376323; 0.883354356; 0.9513968363; 0.9727698268; 0.5315570949; 0.9599239266; 1.564976755; 0.7232002879; 1.084139442; 1.220914517; 0.3553085946; 1.112549824; 1.989443553; 0.5752307543; 1.156682549; 0.7886670467; 0.02050745923; 1.532060208; 1.18789591; 1.408946777; 1.038812004; 1.724679503; 1.671565045; 1.266831442; 1.363611654; 1.705819067; 0.5772366328; 0.4503488498; 1.496891481; 0.9831877282; 0.3845460366; 0.8253240671; 1.298969969; 0.4265904553; 0.9303696876; 0.445003361; 0.753175816; 0.6143534043; 1.059982235; 0.7143206784; 0.2233328038; 1.005178481; 0.7697392436; 0.5904948577; 0.5127953044; 0.6467346747; 0.7929387604; -0.1501790761; 0.8750780903; 0.941704369; 1.37941579; 0.4739006145; 1.998886344; 1.219428519; 0.06270791476; 1.097739804; 0.7584232803; 1.042177217; 1.166561247; 1.502357164; 1.171525776; 0.1528807432; 0.2289389756; 1.36208422; 0.3714421124; 1.299571092; 1.171553369; 1.317807265; 1.616662281; 1.724223246; 1.059580642; 1.270520918; -0.1827677907; 1.938593232; 1.420362143; 1.888357595; 0.7851629936; 0.7080554899; 0.7747215818; 1.403719877; 0.5765950249; 1.275206565; 0.6292054813; 1.525562798; 0.6224640457; 0.8524078517; 0.7646595627; 0.6799834691; 0.773111053; ...]
==F# ==
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() ]
Factor
USING: random ;
1000 [ 1.0 0.5 normal-random-float ] replicate
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
endFantom
Two solutions. The first uses Fantom's random-number generator, which produces a uniform distribution. So, convert to a normal distribution using a formula:
class Main
{
static const Float PI := 0.0f.acos * 2 // we need to precompute PI
static Float randomNormal ()
{
return (Float.random * PI * 2).cos * (Float.random.log * -2).sqrt
}
public static Void main ()
{
mean := 1.0f
sd := 0.5f
Float[] values := [,] // this is the collection to fill with random numbers
1000.times { values.add (randomNormal * sd + mean) }
}
}The second calls out to Java's Gaussian random-number generator:
using [java] java.util::Random
class Main
{
Random generator := Random()
Float randomNormal ()
{
return generator.nextGaussian
}
public static Void main ()
{
rnd := Main() // create an instance of Main class, which holds the generator
mean := 1.0f
sd := 0.5f
Float[] values := [,] // this is the collection to fill with random numbers
1000.times { values.add (rnd.randomNormal * sd + mean) }
}
}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
For newer versions of gforth (tested on 0.7.3), it seems you need to use HERE SEED ! instead of HERE TO SEED, because SEED has been made a variable instead of a value.
rnd rnd dabs d>f
is necessary, but surprising and definitely not well documented / perhaps not compliant.
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
- Output:
Mean = 0.995112 Standard Deviation = 0.503373
Free Pascal
Free Pascal provides the randg function in the RTL math unit that produces Gaussian-distributed random numbers with the Box-Müller algorithm.
function randg(mean,stddev: float): float;
Frink
a = new array[[1000], {|x| randomGaussian[1, 0.5]}]FutureBasic
Note: To generate the random number, rather than using FB's native "rnd" function, this code wraps C code into the RandomZeroToOne function.
window 1
local fn RandomZeroToOne as double
double result
cln result = (double)( (rand() % 100000 ) * 0.00001 );
end fn = result
local fn RandomGaussian as double
double r = fn RandomZeroToOne
end fn = 1 + .5 * ( sqr( -2 * log(r) ) * cos( 2 * pi * r ) )
long i
double mean, std, a(1000)
for i = 1 to 1000
a(i) = fn RandomGaussian
mean += a(i)
next
mean = mean / 1000
for i = 1 to 1000
std += ( a(i) - mean )^2
next
std = std / 1000
print " Average: "; mean
print "Standard Deviation: "; std
HandleEvents- Output:
Average: 1.053724951604593 Standard Deviation: 0.2897370762627166
Go
This solution uses math/rand package in the standard library. See also though the subrepository rand package at https://godoc.org/golang.org/x/exp/rand, which also has a NormFloat64 and has a rand source with a number of advantages over the one in standard library.
package main
import (
"fmt"
"math"
"math/rand"
"strings"
"time"
)
const mean = 1.0
const stdv = .5
const n = 1000
func main() {
var list [n]float64
rand.Seed(time.Now().UnixNano())
for i := range list {
list[i] = mean + stdv*rand.NormFloat64()
}
// show computed mean and stdv of list
var s, sq float64
for _, v := range list {
s += v
}
cm := s / n
for _, v := range list {
d := v - cm
sq += d * d
}
fmt.Printf("mean %.3f, stdv %.3f\n", cm, math.Sqrt(sq/(n-1)))
// show histogram by hdiv divisions per stdv over +/-hrange stdv
const hdiv = 3
const hrange = 2
var h [1 + 2*hrange*hdiv]int
for _, v := range list {
bin := hrange*hdiv + int(math.Floor((v-mean)/stdv*hdiv+.5))
if bin >= 0 && bin < len(h) {
h[bin]++
}
}
const hscale = 10
for _, c := range h {
fmt.Println(strings.Repeat("*", (c+hscale/2)/hscale))
}
}
- Output:
mean 0.995, stdv 0.503 ** **** ****** ******** ************ ************ ************* ************ ********** ******** ***** *** **
Groovy
rnd = new Random()
result = (1..1000).inject([]) { r, i -> r << rnd.nextGaussian() }
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
Or using Data.Random from random-fu package:
replicateM 1000 $ normal 1 0.5
To print them:
import Data.Random
import Control.Monad
thousandRandomNumbers :: RVar [Double]
thousandRandomNumbers = replicateM 1000 $ normal 1 0.5
main = do
x <- sample thousandRandomNumbers
print x
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)) + mIcon 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
Note that Unicon randomly seeds it's generator.
IDL
result = 1.0 + 0.5*randomn(seed,1000)
J
Solution:
urand=: ?@$ 0:
zrand=: (2 o. 2p1 * urand) * [: %: _2 * [: ^. urand
1 + 0.5 * zrand 100
Alternative Solution:
Using the normal script from the stats/distribs addon.
require 'stats/distribs/normal'
1 0.5 rnorm 1000
1.44868803 1.21548637 0.812460657 1.54295452 1.2470606 ...
Java
double[] list = new double[1000];
double mean = 1.0, std = 0.5;
Random rng = new Random();
for(int i = 0;i<list.length;i++) {
list[i] = mean + std * rng.nextGaussian();
}
JavaScript
function randomNormal() {
return Math.cos(2 * Math.PI * Math.random()) * Math.sqrt(-2 * Math.log(Math.random()))
}
var a = []
for (var i=0; i < 1000; i++){
a[i] = randomNormal() / 2 + 1
}
jq
Since jq is a purely functional language, it is convenient to define the pseudo-random number generator functions as filters whose inputs and outputs are arrays containing a "seed".
The following uses the same pseudo-random number generator as the Microsoft C Runtime (see Linear congruential generator).
'A Pseudo-Random Number Generator'
# 15-bit integers generated using the same formula as rand() from the Microsoft C Runtime.
# The random numbers are in [0 -- 32767] inclusive.
# Input: an array of length at least 2 interpreted as [count, state, ...]
# Output: [count+1, newstate, r] where r is the next pseudo-random number.
def next_rand_Microsoft:
.[0] as $count | .[1] as $state
| ( (214013 * $state) + 2531011) % 2147483648 # mod 2^31
| [$count+1 , ., (. / 65536 | floor) ] ;'Box-Muller Method'
# Generate a single number following the normal distribution with mean 0, variance 1,
# using the Box-Muller method: X = sqrt(-2 ln U) * cos(2 pi V) where U and V are uniform on [0,1].
# Input: [n, state]
# Output [n+1, nextstate, r]
def next_rand_normal:
def u: next_rand_Microsoft | .[2] /= 32767;
u as $u1
| ($u1 | u) as $u2
| ((( (8*(1|atan)) * $u1[2]) | cos)
* ((-2 * (($u2[2]) | log)) | sqrt)) as $r
| [ (.[0]+1), $u2[1], $r] ;
# Generate "count" arrays, each containing a random normal variate with the given mean and standard deviation.
# Input: [count, state]
# Output: [updatedcount, updatedstate, rnv]
# where "state" is a seed and "updatedstate" can be used as a seed.
def random_normal_variate(mean; sd; count):
next_rand_normal
| recurse( if .[0] < count then next_rand_normal else empty end)
| .[2] = (.[2] * sd) + mean;Example The task can be completed using: [0,1] | random_normal_variate(1; 0.5; 1000) | .[2]
We show just the sample average and standard deviation:
def summary:
length as $l | add as $sum | ($sum/$l) as $a
| reduce .[] as $x (0; . + ( ($x - $a) | .*. ))
| [ $a, (./$l | sqrt)] ;
[ [0,1] | random_normal_variate(1; 0.5; 1000) | .[2] ] | summary- Output:
$ jq -n -c -f Random_numbers.jq [0.9932830741018853,0.4977760644490579]
Julia
Julia's standard library provides a randn function to generate normally distributed random numbers (with mean 0 and standard deviation 0.5, which can be easily rescaled to any desired values):
randn(1000) * 0.5 + 1
Kotlin
// version 1.0.6
import java.util.Random
fun main(args: Array<String>) {
val r = Random()
val da = DoubleArray(1000)
for (i in 0 until 1000) da[i] = 1.0 + 0.5 * r.nextGaussian()
// now check actual mean and SD
val mean = da.average()
val sd = Math.sqrt(da.map { (it - mean) * (it - mean) }.average())
println("Mean is $mean")
println("S.D. is $sd")
}
Sample output:
- Output:
Mean is 1.0071784073168768 S.D. is 0.48567118114896807
LabVIEW
Lingo
-- Returns a random float value in range 0..1
on randf ()
n = random(the maxinteger)-1
return n / float(the maxinteger-1)
endnormal = []
repeat with i = 1 to 1000
normal.add(1 + sqrt(-2 * log(randf())) * cos(2 * PI * randf()) / 2)
end repeatLobster
Uses built-in rnd_gaussian
let mean = 1.0
let stdv = 0.5
let count = 1000
// stats computes a running mean and variance
// See Knuth TAOCP vol 2, 3rd edition, page 232
def stats(xs: [float]) -> float, float: // variance, mean
var M = xs[0]
var S = 0.0
var n = 1.0
for(xs.length - 1) i:
let x = xs[i + 1]
n = n + 1.0
let mm = (x - M)
M += mm / n
S += mm * (x - M)
return (if n > 0.0: S / n else: 0.0), M
def test_random_normal() -> [float]:
rnd_seed(floor(seconds_elapsed() * 1000000))
let r = vector_reserve(typeof return, count)
for (count):
r.push(rnd_gaussian() * stdv + mean)
let cvar, cmean = stats(r)
let cstdv = sqrt(cvar)
print concat_string(["Mean: ", string(cmean), ", Std.Deviation: ", string(cstdv)], "")
test_random_normal()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.
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 ?] []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
M2000 Interpreter
M2000 use a Wichmann - Hill Pseudo Random Number Generator.
Module CheckIt {
Function StdDev (A()) {
\\ A() has a copy of values
N=Len(A())
if N<1 then Error "Empty Array"
M=Each(A())
k=0
\\ make sum, dev same type as A(k)
sum=A(k)-A(k)
dev=sum
\\ find mean
While M {
sum+=Array(M)
}
Mean=sum/N
\\ make a pointet to A()
P=A()
\\ subtruct from each item
P-=Mean
M=Each(P)
While M {
dev+=Array(M)*Array(M)
}
\\ as pointer to arrray
=(if(dev>0->Sqrt(dev/N), 0), Mean)
}
Function randomNormal {
\\ by default all numbers are double
\\ cos() get degrees
=1+Cos(360 * rnd) * Sqrt(-2 * Ln(rnd)) /2
}
\\ fill array calling randomNormal() for each item
Dim A(1000)<<randomNormal()
\\ we can pass a pointer to array and place it to stack of values
DisplayMeanAndStdDeviation(A()) ' mean ~ 1 std deviation ~0.5
\\ check M2000 rnd only
Dim B(1000)<<rnd
DisplayMeanAndStdDeviation(B()) ' mean ~ 0.5 std deviation ~0.28
DisplayMeanAndStdDeviation((0,0,14,14)) ' mean = 7 std deviation = 7
DisplayMeanAndStdDeviation((0,6,8,14)) ' mean = 7 std deviation = 5
DisplayMeanAndStdDeviation((6,6,8,8)) ' mean = 7 std deviation = 1
Sub DisplayMeanAndStdDeviation(A)
\\ push to stack all items of an array (need an array pointer)
Push ! StdDev(A)
\\ read from strack two numbers
Print "Mean is "; Number
Print "Standard Deviation is "; Number
End Sub
}
CheckitMaple
with(Statistics):
Sample(Normal(1, 0.5), 1000);or
1+0.5*ArrayTools[RandomArray](1000,1,distribution=normal);Mathematica /Wolfram Language
Built-in function RandomReal with built-in distribution NormalDistribution as an argument:
RandomReal[NormalDistribution[1, 1/2], 1000]
MATLAB
Native support :
mu = 1; sd = 0.5;
x = randn(1000,1) * sd + mu;
The statistics toolbox provides this function
x = normrnd(mu, sd, [1000,1]);
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.
function randNum = randNorm(mu0,chi2, sz)
radiusSquared = +Inf;
while (radiusSquared >= 1)
u = ( 2 * rand(sz) ) - 1;
v = ( 2 * rand(sz) ) - 1;
radiusSquared = u.^2 + v.^2;
end
scaleFactor = sqrt( ( -2*log(radiusSquared) )./ radiusSquared );
randNum = (v .* scaleFactor .* chi2) + mu0;
end
Output:
>> randNorm(1,.5, [1000,1])
ans =
0.693984121077029
Maxima
load(distrib)$
random_normal(1.0, 0.5, 1000);
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
)Metafont
Metafont has normaldeviate which produces pseudorandom normal distributed numbers with mean 0 and variance one. So the following complete the task:
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
endA 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
MiniScript
randNormal = function(mean=0, stddev=1)
return mean + sqrt(-2 * log(rnd,2.7182818284)) * cos(2*pi*rnd) * stddev
end function
x = []
for i in range(1,1000)
x.push randNormal(1, 0.5)
end for
Mirah
import java.util.Random
list = double[999]
mean = 1.0
std = 0.5
rng = Random.new
0.upto(998) do | i |
list[i] = mean + std * rng.nextGaussian
endМК-61/52
П7 <-> П8 1/x П6 ИП6 П9 СЧ П6 1/x
ln ИП8 * 2 * КвКор ИП9 2 * пи
* sin * ИП7 + С/П БП 05
Input: РY - variance, РX - expectation.
Or:
3 10^x П0 ПП 13 2 / 1 + С/П L0 03 С/П
СЧ lg 2 /-/ * КвКор 2 пи ^ СЧ * * cos * В/О
to generate 1000 numbers with a mean of 1.0 and a standard deviation of 0.5.
Modula-3
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.
Nanoquery
list = {0} * 1000
mean = 1.0; std = 0.5
rng = new(Nanoquery.Util.Random)
for i in range(0, len(list) - 1)
list[i] = mean + std * rng.getGaussian()
endNetRexx
/* NetRexx */
options replace format comments java crossref symbols nobinary
import java.math.BigDecimal
import java.math.MathContext
-- prologue
numeric digits 20
-- get input, set defaults
parse arg dp mu sigma ec .
if mu = '' | mu = '.' then mean = 1.0; else mean = mu
if sigma = '' | sigma = '.' then stdDeviation = 0.5; else stdDeviation = sigma
if dp = '' | dp = '.' then displayPrecision = 1; else displayPrecision = dp
if ec = '' | ec = '.' then elements = 1000; else elements = ec
-- set up
RNG = Random()
numberList = java.util.List
numberList = ArrayList()
-- generate list of random numbers
loop for elements
rn = mean + stdDeviation * RNG.nextGaussian()
numberList.add(BigDecimal(rn, MathContext.DECIMAL128))
end
-- report
say "Mean: " mean
say "Standard Deviation:" stdDeviation
say "Precision: " displayPrecision
say
drawBellCurve(numberList, displayPrecision)
return
-- -----------------------------------------------------------------------------
method drawBellCurve(numberList = java.util.List, precision) static
Collections.sort(numberList)
val = BigDecimal
lastN = ''
nextN = ''
loop val over numberList
nextN = Rexx(val.toPlainString()).format(5, precision)
select
when lastN = '' then nop
when lastN \= nextN then say lastN
otherwise nop
end
say '*\-'
lastN = nextN
end val
say lastN
return
- Output:
Mean: 1.0 Standard Deviation: 0.5 Precision: 1 * 2.7 ** 2.5 * 2.4 *** 2.3 ***** 2.2 ******* 2.1 ************* 2.0 ************* 1.9 ***************************** 1.8 ************************* 1.7 ************************************* 1.6 ****************************************************** 1.5 ******************************************** 1.4 ******************************************************************** 1.3 ***************************************************************** 1.2 ************************************************************************** 1.1 ********************************************************************************************* 1.0 ************************************************************* 0.9 ********************************************************************** 0.8 ************************************************************** 0.7 *********************************************************************** 0.6 ************************************************************** 0.5 ****************************************** 0.4 ******************************* 0.3 *************************** 0.2 *************** 0.1 ********* 0.0 ****** -0.1 *** -0.2 *** -0.3 * -0.4 * -0.6 ** -0.7
NewLISP
(normal 1 .5 1000)
Nim
import random, stats, strformat
var rs: RunningStat
randomize()
for _ in 1..5:
for _ in 1..1000: rs.push gauss(1.0, 0.5)
echo &"mean: {rs.mean:.5f} stdDev: {rs.standardDeviation:.5f}"
- Output:
mean: 1.01294 stdDev: 0.49692 mean: 1.00262 stdDev: 0.50028 mean: 0.99878 stdDev: 0.49662 mean: 0.99830 stdDev: 0.49820 mean: 1.00658 stdDev: 0.49703
Nu
def 'seq rnd-nomal' [mean sdev] {
0.. | each { random float (-1)..1 } | window 2
| each { [($in.0 ** 2 + $in.1 ** 2) $in.1] }
| where { 0 < $in.0 and $in.0 < 1 }
| each { do {|q r| -2 * ($q | math ln) / $q | math sqrt | $sdev * $in * $r + $mean } ...$in }
}
let rand = seq rnd-nomal 1.0 0.5 | take 1000
$'mean: ($rand | math avg), standard deviation: ($rand | math stddev)'- Output (example):
mean: 1.0156044868464669, standard deviation: 0.5010622108454892
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();
}
}
}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 ());;
Octave
p = normrnd(1.0, 0.5, 1000, 1);
disp(mean(p));
disp(sqrt(sum((p - mean(p)).^2)/numel(p)));
- Output:
1.0209 0.51048
ooRexx
version 1
/*REXX pgm gens 1,000 normally distributed #s: mean=1, standard dev.=0.5*/
pi=RxCalcPi() /* get value of pi */
Parse Arg n seed . /* allow specification of N & seed*/
If n==''|n==',' Then
n=1000 /* N is the size of the array. */
If seed\=='' Then
Call random,,seed /* use seed for repeatable RANDOM#*/
mean=1 /* desired new mean (arith. avg.) */
sd=1/2 /* desired new standard deviation.*/
Do g=1 For n /* generate N uniform random nums.*/
n.g=random(0,1e5)/1e5 /* REXX gens uniform rand integers*/
End
Say ' old mean=' mean()
Say 'old standard deviation=' stddev()
Say
Do j=1 To n-1 By 2
m=j+1
/*use Box-Muller method */
_=sd*RxCalcPower(-2*RxCalcLog(n.j),.5)*RxCalcCos(2*pi*n.m,,'R')+mean
n.m=sd*RxCalcpower(-2*RxCalcLog(n.j),.5)*RxCalcSin(2*pi*n.m,,'R')+,
mean /* rand # must be 0???1. */
n.j=_
End /* j */
Say ' new mean=' mean()
Say 'new standard deviation=' stddev()
Exit
mean:
_=0
Do k=1 For n
_=_+n.k
End
Return _/n
stddev:
_avg=mean()
_=0
Do k=1 For n
_=_+(n.k-_avg)**2
End
Return RxCalcPower(_/n,.5)
:: requires rxmath library
- Output:
old mean= 0.49830002
old standard deviation= 0.283199568
new mean= 1.00377404
new standard deviation= 0.501444536
version 2
Using the nice function names in the algorithm.
/*REXX pgm gens 1,000 normally distributed #s: mean=1, standard dev.=0.5*/
pi=RxCalcPi() /* get value of pi */
Parse Arg n seed . /* allow specification of N & seed*/
If n==''|n==',' Then
n=1000 /* N is the size of the array. */
If seed\=='' Then
Call random,,seed /* use seed for repeatable RANDOM#*/
mean=1 /* desired new mean (arith. avg.) */
sd=1/2 /* desired new standard deviation.*/
Do g=1 For n /* generate N uniform random nums.*/
n.g=random(0,1e5)/1e5 /* REXX gens uniform rand integers*/
End
Say ' old mean=' mean()
Say 'old standard deviation=' stddev()
Say
Do j=1 To n-1 By 2
m=j+1
/*use Box-Muller method */
_=sd*sqrt(-2*ln(n.j))*cos(2*pi*n.m)+mean
n.m=sd*sqrt(-2*ln(n.j))*sin(2*pi*n.m)+mean
n.j=_
End
Say ' new mean=' mean()
Say 'new standard deviation=' stddev()
Exit
mean:
_=0
Do k=1 For n
_=_+n.k
End
Return _/n
stddev:
_avg=mean()
_=0
Do k=1 For n
_=_+(n.k-_avg)**2
End
Return sqrt(_/n)
sqrt: Return RxCalcSqrt(arg(1))
ln: Return RxCalcLog(arg(1))
cos: Return RxCalcCos(arg(1),,'R')
sin: Return RxCalcSin(arg(1),,'R')
:: requires rxmath library
PARI/GP
rnormal()={
my(pr=32*ceil(default(realprecision)*log(10)/log(4294967296)),u1=random(2^pr)*1.>>pr,u2=random(2^pr)*1.>>pr);
sqrt(-2*log(u1))*cos(2*Pi*u2) \\ in previous version "u1" instead of "u2" was used --> has given crap distribution
\\ Could easily be extended with a second normal at very little cost.
};
vector(1000,unused,rnormal()/2+1)Pascal
The following function calculates Gaussian-distributed random numbers with the Box-Müller algorithm:
function rnorm (mean, sd: real): real;
{Calculates Gaussian random numbers according to the Box-Müller approach}
var
u1, u2: real;
begin
u1 := random;
u2 := random;
rnorm := mean * abs(1 + sqrt(-2 * (ln(u1))) * cos(2 * pi * u2) * sd);
/* error !?! Shouldn't it be "mean +" instead of "mean *" ? */
end;
Delphi and Free Pascal support implement a randg function that delivers Gaussian-distributed random numbers.
PascalABC.NET
const
n = 1000;
function randnormal(n: integer; mean, sd: real): array of real;
begin
var uniform1 := ArrRandomReal(n, 0, 1, 10);
var uniform2 := ArrRandomReal(n, 0, 1, 10);
result := uniform1.Zip(uniform2, (x, y) -> Cos(2 * PI * x) * Sqrt(-2 * Ln(y)) * sd + mean).ToArray;
end;
begin
var numbers := randnormal(n, 1, 0.5);
var mean := numbers.Average;
var std := numbers.Sum(x -> Sqr(x - mean) / n).Sqrt;
println('mean:', mean, 'stddev:', std);
end.
- Output:
mean: 1.00038329067095 stddev: 0.506417472572872
Perl
my $PI = 2 * atan2 1, 0;
my @nums = map {
1 + 0.5 * sqrt(-2 * log rand) * cos(2 * $PI * rand)
} 1..1000;
Phix
function RandomNormal() return sqrt(-2*log(rnd())) * cos(2*PI*rnd()) end function sequence s = repeat(0,1000) for i=1 to length(s) do s[i] = 1 + 0.5 * RandomNormal() end for
Phixmonti
include ..\Utilitys.pmt
def RandomNormal
drop rand log -2 * sqrt 2 pi * rand * cos * 0.5 * 1 +
enddef
1000 var n
0 n repeat
getid RandomNormal map
dup
sum n / var mean
"Mean: " print mean print nl
0 swap n for
get mean - 2 power rot + swap
endfor
swap n / sqrt "Standard deviation: " print printPHP
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);
}
Picat
main =>
_ = random2(), % random seed
G = [gaussian_dist(1,0.5) : _ in 1..1000],
println(first_10=G[1..10]),
println([mean=avg(G),stdev=stdev(G)]),
nl.
% Gaussian (Normal) distribution, Box-Muller algorithm
gaussian01() = Y =>
U = frand(0,1),
V = frand(0,1),
Y = sqrt(-2*log(U))*sin(2*math.pi*V).
gaussian_dist(Mean,Stdev) = Mean + (gaussian01() * Stdev).
% Variance of Xs
variance(Xs) = Variance =>
Mu = avg(Xs),
N = Xs.len,
Variance = sum([ (X-Mu)**2 : X in Xs ]) / N.
% Standard deviation
stdev(Xs) = sqrt(variance(Xs)).- Output:
first_10 = [1.639965415776091,0.705425965005482,0.981532402477848,0.309148743347499,1.252800181962738,0.098829881195179,0.74888084504147,0.181494956495445,1.304931340021904,0.595939453660087] [mean = 0.99223677282248,stdev = 0.510336641737154]
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) " ") ) )- Output:
1.500334 1.212931 1.095283 0.433122 0.459116 1.302446 0.402477
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;- Output:
Mean = 1.0125630677913652 Standard Deviation = 0.5067289784535284 3 runs with different seeds to random(): Mean = 1.0008390411168471 Standard Deviation = 0.5095810511317908 Mean = 0.9754351286894838 Standard Deviation = 0.4804376530558166 Mean = 1.0177411222687990 Standard Deviation = 0.5165899662493400
PL/SQL
DECLARE
--The desired collection
type t_coll is table of number index by binary_integer;
l_coll t_coll;
c_max pls_integer := 1000;
BEGIN
FOR l_counter IN 1 .. c_max
LOOP
-- dbms_random.normal delivers normal distributed random numbers with a mean of 0 and a variance of 1
-- We just adjust the values and get the desired result:
l_coll(l_counter) := DBMS_RANDOM.normal * 0.5 + 1;
DBMS_OUTPUT.put_line (l_coll(l_counter));
END LOOP;
END;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;PowerShell
Equation adapted from Liberty BASIC
function Get-RandomNormal
{
[CmdletBinding()]
Param ( [double]$Mean, [double]$StandardDeviation )
$RandomNormal = $Mean + $StandardDeviation * [math]::Sqrt( -2 * [math]::Log( ( Get-Random -Minimum 0.0 -Maximum 1.0 ) ) ) * [math]::Cos( 2 * [math]::PI * ( Get-Random -Minimum 0.0 -Maximum 1.0 ) )
return $RandomNormal
}
# Standard deviation function for testing
function Get-StandardDeviation
{
[CmdletBinding()]
param ( [double[]]$Numbers )
$Measure = $Numbers | Measure-Object -Average
$PopulationDeviation = 0
ForEach ($Number in $Numbers) { $PopulationDeviation += [math]::Pow( ( $Number - $Measure.Average ), 2 ) }
$StandardDeviation = [math]::Sqrt( $PopulationDeviation / ( $Measure.Count - 1 ) )
return $StandardDeviation
}
# Test
$RandomNormalNumbers = 1..1000 | ForEach { Get-RandomNormal -Mean 1 -StandardDeviation 0.5 }
$Measure = $RandomNormalNumbers | Measure-Object -Average
$Stats = [PSCustomObject]@{
Count = $Measure.Count
Average = $Measure.Average
StandardDeviation = Get-StandardDeviation -Numbers $RandomNormalNumbers
}
$Stats | Format-List
- Output:
Count : 1000 Average : 1.01206560135809 StandardDeviation : 0.489099623426272
Python
- Using random.gauss
>>> import random
>>> values = [random.gauss(1, .5) for i in range(1000)]
>>>
- Quick check of distribution
>>> def quick_check(numbers):
count = len(numbers)
mean = sum(numbers) / count
sdeviation = (sum((i - mean)**2 for i in numbers) / count)**0.5
return mean, sdeviation
>>> quick_check(values)
(1.0140373306786599, 0.49943411329234066)
>>>
Note that the random module in the Python standard library supports a number of statistical distribution methods.
- Alternatively using random.normalvariate
>>> values = [ random.normalvariate(1, 0.5) for i in range(1000)]
>>> quick_check(values)
(0.990099111944864, 0.5029847005836282)
>>>
R
# For reproducibility, set the seed:
set.seed(12345L)
result <- rnorm(1000, mean = 1, sd = 0.5)
Racket
#lang racket
(for/list ([i 1000])
(add1 (* (sqrt (* -2 (log (random)))) (cos (* 2 pi (random))) 0.5)))
Alternative:
#lang racket
(require math/distributions)
(sample (normal-dist 1.0 0.5) 1000)
Raku
(formerly Perl 6)
sub randnorm ($mean, $stddev) {
$mean + $stddev * sqrt(-2 * log rand) * cos(2 * pi * rand)
}
my @nums = randnorm(1, 0.5) xx 1000;
# Checking
say my $mean = @nums R/ [+] @nums;
say my $stddev = sqrt $mean**2 R- @nums R/ [+] @nums X** 2;
Raven
define PI
-1 acos
define rand1
9999999 choose 1 + 10000000.0 /
define randNormal
rand1 PI * 2 * cos
rand1 log -2 * sqrt
*
2 / 1 +
1000 each drop randNormal "%f\n" printQuick Check (on linux with code in file rand.rv)
raven rand.rv | awk '{sum+=$1; sumsq+=$1*$1;} END {print "stdev = " sqrt(sumsq/NR - (sum/NR)**2); print "mean = " sum/NR}'
stdev = 0.497773
mean = 1.01497ReScript
let pi = 4.0 *. atan(1.0)
let random_gaussian = () => {
1.0 +.
sqrt(-2.0 *. log(Random.float(1.0))) *.
cos(2.0 *. pi *. Random.float(1.0))
}
let a = Belt.Array.makeBy(1000, (_) => random_gaussian ())
for i in 0 to 10 {
Js.log(a[i])
}REXX
The REXX language doesn't have any "higher math" functions like SQRT/SIN/COS/LN/LOG/EXP/POW/etc.,
so we hoi polloi REXX programmers have to roll our own.
Programming note: note the range of the random numbers: (0,1]
(that is, random numbers from zero──►unity, excluding zero, including unity).
/*REXX pgm generates 1,000 normally distributed numbers: mean=1, standard deviation=½.*/
numeric digits 20 /*the default decimal digit precision=9*/
parse arg n seed . /*allow specification of N and the seed*/
if n=='' | n=="," then n=1000 /*N: is the size of the array. */
if datatype(seed,'W') then call random ,,seed /*SEED: for repeatable random numbers. */
newMean=1 /*the desired new mean (arithmetic avg)*/
sd=1/2 /*the desired new standard deviation. */
do g=1 for n /*generate N uniform random #'s (0,1].*/
#.g = random(1, 1e5) / 1e5 /*REXX's RANDOM BIF generates integers.*/
end /*g*/ /* [↑] random integers ──► fractions. */
say ' old mean=' mean()
say 'old standard deviation=' stdDev()
call pi; pi2=pi * 2 /*define pi and also 2 * pi. */
say
do j=1 to n-1 by 2; m=j+1 /*step through the iterations by two. */
_=sd * sqrt(ln(#.j) * -2) /*calculate the used-twice expression.*/
#.j=_ * cos(pi2 * #.m) + newMean /*utilize the Box─Muller method. */
#.m=_ * sin(pi2 * #.m) + newMean /*random number must be: (0,1] */
end /*j*/
say ' new mean=' mean()
say 'new standard deviation=' stdDev()
exit /*stick a fork in it, we're all done. */
/*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
mean: _=0; do k=1 for n; _=_ + #.k; end; return _/n
stdDev: _avg=mean(); _=0; do k=1 for n; _=_ + (#.k - _avg)**2; end; return sqrt(_/n)
e: e =2.7182818284590452353602874713526624977572470936999595749669676277240766303535; return e /*digs overkill*/
pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862; return pi /* " " */
r2r: return arg(1) // (pi() * 2) /*normalize ang*/
sin: procedure; parse arg x;x=r2r(x);numeric fuzz min(5,digits()-3);if abs(x)=pi then return 0;return .sincos(x,x,1)
.sincos:parse arg z,_,i; x=x*x; p=z; do k=2 by 2; _=-_*x/(k*(k+i)); z=z+_; if z=p then leave; p=z; end; return z
/*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
ln: procedure; parse arg x,f; call e; ig= x>1.5; is=1 - 2 * (ig\==1); ii=0; xx=x
do while ig&xx>1.5|\ig&xx<.5;_=e;do k=-1;iz=xx*_**-is;if k>=0&(ig&iz<1|\ig&iz>.5) then leave;_=_*_;izz=iz;end
xx=izz;ii=ii+is*2**k;end;x=x*e**-ii-1;z=0;_=-1;p=z;do k=1;_=-_*x;z=z+_/k;if z=p then leave;p=z;end; return z+ii
/*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
cos: procedure; parse arg x; x=r2r(x); a=abs(x); hpi=pi * .5
numeric fuzz min(6, digits() - 3); if a=pi then return -1
if a=hpi | a=hpi*3 then return 0; if a=pi/3 then return .5
if a=pi * 2/3 then return -.5; return .sinCos(1,1,-1)
/*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); numeric digits; h=d+6
numeric form; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g * .5'e'_ %2
m.=9; do j=0 while h>9; m.j=h; h=h%2 + 1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/
numeric digits d; return g/1
output when using the default inputs:
old mean= 0.5015724
old standard deviation= 0.28652466389342471402
new mean= 0.98807025356443262689
new standard deviation= 0.50002924192766720838
Ring
for i = 1 to 10
see random(i) + nl
next iRPL
≪ RAND LN NEG 2 * √ RAND 2 * π * COS * →NUM 2 / 1 + ≫ 'RANDN' STO ≪ CL∑ 1 1000 START RANDN ∑+ NEXT MEAN PSDEV ≫ 'TASK' STO
- Output:
1: .990779804949 2: .487204045227
The collection is stored in a predefined array named ∑DAT, which is automatically created/updated when using the ∑+ instruction and remains available until the user decides to purge it, typically by calling the CL∑ command.
Ruby
Array.new(1000) { 1 + Math.sqrt(-2 * Math.log(rand)) * Math.cos(2 * Math::PI * rand) }
Rust
Using a for-loop:
extern crate rand;
use rand::distributions::{Normal, IndependentSample};
fn main() {
let mut rands = [0.0; 1000];
let normal = Normal::new(1.0, 0.5);
let mut rng = rand::thread_rng();
for num in rands.iter_mut() {
*num = normal.ind_sample(&mut rng);
}
}
Using iterators:
extern crate rand;
use rand::distributions::{Normal, IndependentSample};
fn main() {
let rands: Vec<_> = {
let normal = Normal::new(1.0, 0.5);
let mut rng = rand::thread_rng();
(0..1000).map(|_| normal.ind_sample(&mut rng)).collect()
};
}
SAS
/* Generate 1000 random numbers with mean 1 and standard deviation 0.5.
SAS version 9.2 was used to create this code.*/
data norm1000;
call streaminit(123456);
/* Set the starting point, so we can replicate results.
If you want different results each time, comment the above line. */
do i=1 to 1000;
r=rand('normal',1,0.5);
output;
end;
run;
Results:
The MEANS Procedure
Analysis Variable : r
Mean Std Dev
----------------------------
0.9907408 0.4844051
----------------------------
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;Scala
One liner
List.fill(1000)(1.0 + 0.5 * scala.util.Random.nextGaussian)
Academic
object RandomNumbers extends App {
val distribution: LazyList[Double] = {
def randomNormal: Double = 1.0 + 0.5 * scala.util.Random.nextGaussian
def normalDistribution(a: Double): LazyList[Double] = a #:: normalDistribution(randomNormal)
normalDistribution(randomNormal)
}
/*
* Let's test it
*/
def calcAvgAndStddev[T](ts: Iterable[T])(implicit num: Fractional[T]): (T, Double) = {
val mean: T =
num.div(ts.sum, num.fromInt(ts.size)) // Leaving with type of function T
// Root of mean diffs
val stdDev = Math.sqrt(ts.map { x =>
val diff = num.toDouble(num.minus(x, mean))
diff * diff
}.sum / ts.size)
(mean, stdDev)
}
println(calcAvgAndStddev(distribution.take(1000))) // e.g. (1.0061433267806525,0.5291834867560893)
}
Scheme
; linear congruential generator given in C99 section 7.20.2.1
(define ((c-rand seed)) (set! seed (remainder (+ (* 1103515245 seed) 12345) 2147483648)) (quotient seed 65536))
; uniform real numbers in open interval (0, 1)
(define (unif-rand seed) (let ((r (c-rand seed))) (lambda () (/ (+ (r) 1) 32769.0))))
; Box-Muller method to generate normal distribution
(define (normal-rand unif m s)
(let ((? #t) (! 0.0) (twopi (* 2.0 (acos -1.0))))
(lambda ()
(set! ? (not ?))
(if ? !
(let ((a (sqrt (* -2.0 (log (unif))))) (b (* twopi (unif))))
(set! ! (+ m (* s a (sin b))))
(+ m (* s a (cos b))))))))
(define rnorm (normal-rand (unif-rand 0) 1.0 0.5))
; auxiliary function to get a list of 'n random numbers from generator 'r
(define (rand-list r n) = (if (zero? n) '() (cons (r) (rand-list r (- n 1)))))
(define v (rand-list rnorm 1000))
v
#|
(-0.27965824722565835
-0.8870860825789542
0.6499618744638194
0.31336141955110863
...
0.5648743998193049
0.8282656735558756
0.6399951934564637
0.7699535302478072)
|#
; check mean and standard deviation
(define (mean-sdev v)
(let loop ((v v) (a 0) (b 0) (n 0))
(if (null? v)
(let ((mean (/ a n)))
(list mean (sqrt (/ (- b (* n mean mean)) (- n 1)))))
(let ((x (car v)))
(loop (cdr v) (+ a x) (+ b (* x x)) (+ n 1))))))
(mean-sdev v)
; (0.9562156817697293 0.5097087109575911)
Seed7
$ include "seed7_05.s7i";
include "float.s7i";
include "math.s7i";
const func float: frand is func # Uniform distribution, (0..1]
result
var float: frand is 0.0;
begin
repeat
frand := rand(0.0, 1.0);
until frand <> 0.0;
end func;
const func float: randomNormal is # Normal distribution, centered on 0, std dev 1
return sqrt(-2.0 * log(frand)) * cos(2.0 * PI * frand);
const proc: main is func
local
var integer: i is 0;
var array float: rands is 1000 times 0.0;
begin
for i range 1 to length(rands) do
rands[i] := 1.0 + 0.5 * randomNormal;
end for;
end func;Sidef
var arr = 1000.of { 1 + (0.5 * sqrt(-2 * 1.rand.log) * cos(Num.tau * 1.rand)) }
arr.each { .say }
Standard ML
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.
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 ());
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.
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 ());
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.
The SML Basis Library does not provide a routine for uniform deviate generation, and PolyML does not have one. Using a routine from "Monte Carlo" by Fishman (Springer), in the function uniformdeviate, and avoiding the slow IntInf's:
val urandomlist = fn seed => fn n =>
let
val uniformdeviate = fn seed =>
let
val in31m = (Real.fromInt o Int32.toInt ) (getOpt (Int32.maxInt,0) );
val in31 = in31m +1.0;
val s1 = 41160.0;
val s2 = 950665216.0;
val v = Real.realFloor seed;
val val1 = v*s1;
val val2 = v*s2;
val next1 = Real.fromLargeInt (Real.toLargeInt IEEEReal.TO_NEGINF (val1/in31)) ;
val next2 = Real.rem(Real.realFloor(val2/in31) , in31m );
val valt = val1+val2 - (next1+next2)*in31m;
val nextt = Real.realFloor(valt/in31m);
val valt = valt - nextt*in31m;
in
(valt/in31m,valt)
end;
val store = ref (0.0,0.0);
val rec u = fn S => fn 0 => [] | n=> (store:=uniformdeviate S; (#1 (!store)):: (u (#2 (!store)) (n-1))) ;
in
u seed n
end;
local
open Math
in
val bmconv = fn urand => fn vrand => 1.0+0.5*(sqrt(~2.0*ln urand)*cos (2.0*pi*vrand) )
end;
val rec makeNormals = fn once => fn u::v::[] => [once u v] |
u::v::rm => (once u v )::(makeNormals once rm );
val anyrealseed=1009.0 ;
makeNormals bmconv (urandomlist anyrealseed 2000);
Stata
clear all
set obs 1000
gen x=rnormal(1,0.5)
Mata
a = rnormal(1000,1,1,0.5)
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()}]
}TorqueScript
for (%i = 0; %i < 1000; %i++)
%list[%i] = 1 + mSqrt(-2 * mLog(getRandom())) * mCos(2 * $pi * getRandom());Uiua
# Generate normal distribution with mean = 1, sd = 0.5
Gauss ← (×(∿+η××2π⚂) (√ׯ2ₙe⚂))
[⍥(+1×0.5Gauss)1000] # -> mean = 1, sd = 0.5
Mean ← ÷⧻⟜/+
Sd ← √÷⊃(⋅⧻|/+×.-)Mean.
⊸⊃Sd Mean- Output:
[0.5549431582333864 0.6318745755493541 0.243308532933335 ...etc...] 1.026433943526625 0.4953224276300106
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.
#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>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)>
Visual FoxPro
LOCAL i As Integer, m As Double, n As Integer, sd As Double
py = PI()
SET TALK OFF
SET DECIMALS TO 6
CREATE CURSOR gdev (deviate B(6))
RAND(-1)
n = 1000
m = 1
sd = 0.5
CLEAR
FOR i = 1 TO n
INSERT INTO gdev VALUES (GaussDev(m, 1/sd))
ENDFOR
CALCULATE AVG(deviate), STD(deviate) TO m, sd
? "Mean", m, "Std Dev", sd
SET TALK ON
SET DECIMALS TO
FUNCTION GaussDev(mean As Double, sdev As Double) As Double
LOCAL z As Double
z = SQRT(-2*LOG(RAND()))*COS(py*RAND())
IF sdev # 0
z = mean + z/sdev
ENDIF
RETURN z
ENDFUNCV (Vlang)
import crypto.rand
fn main() {
mut nums := []u64{}
for _ in 0..1000 {
nums << rand.int_u64(10000) or {0} // returns random unsigned 64-bit integer from real OS source of entropy
}
println(nums)
}Wren
import "random" for Random
var rand = Random.new()
var randNormal = Fn.new { (-2 * rand.float().log).sqrt * (2 * Num.pi * rand.float()).cos }
var stdDev = Fn.new { |a, m|
var c = a.count
return ((a.reduce(0) { |acc, x| acc + x*x } - m*m*c) / c).sqrt
}
var n = 1000
var numbers = List.filled(n, 0)
var mu = 1
var sigma = 0.5
var sum = 0
for (i in 0...n) {
numbers[i] = mu + sigma*randNormal.call()
sum = sum + numbers[i]
}
var mean = sum / n
System.print("Actual mean : %(mean)")
System.print("Actual std dev: %(stdDev.call(numbers, mean))")- Output:
Sample run:
Actual mean : 1.0053988699746 Actual std dev: 0.4961645117026
XPL0
define PI = 3.14159265358979323846;
func real DRand; \Uniform distribution, [0..1]
return float(Ran(1_000_000)) / 1e6;
func real RandomNormal; \Normal distribution, centered on 0, std dev 1
return sqrt(-2.*Log(DRand)) * Cos(2.*PI*DRand);
int I;
real Rands(1000);
for I:= 0 to 1000-1 do
Rands(I):= 1.0 + 0.5*RandomNormalYorick
Returns array of count random numbers with mean 0 and standard deviation 1.
func random_normal(count) {
return sqrt(-2*log(random(count))) * cos(2*pi*random(count));
}Example of basic use:
> nums = random_normal(1000); // create an array 1000 random numbers > nums(avg); // show the mean 0.00901216 > nums(rms); // show the standard deviation 0.990265
Example with a mean of 1.0 and a standard deviation of 0.5:
> nums = random_normal(1000) * 0.5 + 1; > nums(avg); 1.00612 > nums(rms); 0.496853
zkl
fcn mkRand(mean,sd){ //normally distributed random w/mean & standard deviation
pi:=(0.0).pi; // using the Box–Muller transform
rz1:=fcn{1.0-(0.0).random(1)} // from [0,1) to (0,1]
return('wrap(){((-2.0*rz1().log()).sqrt() * (2.0*pi*rz1()).cos())*sd + mean })
}This creates a new random number generator, now to use it:
var g=mkRand(1,0.5);
ns:=(0).pump(1000,List,g); // 1000 rands with mean==1 & sd==1/2
mean:=(ns.sum(0.0)/1000); //-->1.00379
// calc sd of list of numbers:
(ns.reduce('wrap(p,n){p+(n-mean).pow(2)},0.0)/1000).sqrt() //-->0.494844Categories:
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