Generate random numbers following a normal distribution in C C

Question

How can I easily generate random numbers following a normal distribution in C or C     I don t want any use of Boost   I know that Knuth talks about this at length but I don t have his books at hand right now

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Have a look on  http   www cplusplus com reference random normal distribution   It s the simplest way to produce normal distributions

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Take a look at what I found   This library uses the Ziggurat algorithm

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1  Graphically intuitive way you can generate Gaussian random numbers is by using something similar to the Monte Carlo method  You would generate a random point in a box around the Gaussian curve using your pseudo-random number generator in C  You can calculate if that point is inside or underneath the Gaussian distribution using the equation of the distribution  If that point is inside the Gaussian distribution  then you have got your Gaussian random number as the x value of the point    This method isn t perfect because technically the Gaussian curve goes on towards infinity  and you couldn t create a box that approaches infinity in the x dimension  But the Guassian curve approaches 0 in the y dimension pretty fast so I wouldn t worry about that  The constraint of the size of your variables in C may be more of a limiting factor to your accuracy   2  Another way would be to use the Central Limit Theorem which states that when independent random variables are added  they form a normal distribution  Keeping this theorem in mind  you can approximate a Gaussian random number by adding a large amount of independent random variables    These methods aren t the most practical  but that is to be expected when you don t want to use a preexisting library  Keep in mind this answer is coming from someone with little or no calculus or statistics experience

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Here s a C   example  based on some of the references  This is quick and dirty  you are better off not re-inventing and using the boost library    include  math h     for RAND  and rand double sampleNormal         double u     double  rand      RAND MAX     2 - 1      double v     double  rand      RAND MAX     2 - 1      double r   u   u   v   v      if  r    0    r  gt  1  return sampleNormal        double c   sqrt -2   log r    r       return u   c      You can use a Q-Q plot to examine the results and see how well it approximates a real normal distribution  rank your samples 1  x  turn the ranks into proportions of total count of x ie  how many samples  get the z-values and plot them  An upwards straight line is the desired result

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C  11  C  11 offers std  normal distribution  which is the way I would go today   C or older C    Here are some solutions in order of ascending complexity    Add 12 uniform random numbers from 0 to 1 and subtract 6  This will match mean and standard deviation of a normal variable  An obvious drawback is that the range is limited to   6     unlike a true normal distribution  The Box-Muller transform  This is listed above  and is relatively simple to implement  If you need very precise samples  however  be aware that the Box-Muller transform combined with some uniform generators suffers from an anomaly called Neave Effect1  For best precision  I suggest drawing uniforms and applying the inverse cumulative normal distribution to arrive at normally distributed variates  Here is a very good algorithm for inverse cumulative normal distributions    1  H  R  Neave     On using the Box-Muller transformation with multiplicative congruential pseudorandom number generators     Applied Statistics  22  92-97  1973

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I created a C   open source project for normally distributed random number generation benchmark    It compares several algorithms  including   Central limit theorem method Box-Muller transform Marsaglia polar method Ziggurat algorithm Inverse transform sampling method  cpp11random uses C  11 std  normal distribution with std  minstd rand  it is actually Box-Muller transform in clang     The results of single-precision  float  version on iMac Corei5-3330S 2 70GHz   clang 6 1  64-bit      For correctness  the program verifies the mean  standard deviation  skewness and kurtosis of the samples  It was found that CLT method by summing 4  8 or 16 uniform numbers do not have good kurtosis as the other methods   Ziggurat algorithm has better performance than the others  However  it does not suitable for SIMD parallelism as it needs table lookup and branches  Box-Muller with SSE2 AVX instruction set is much faster  x1 79  x2 99  than non-SIMD version of ziggurat algorithm   Therefore  I will suggest using Box-Muller for architecture with SIMD instruction sets  and may be ziggurat otherwise     P S  the benchmark uses a simplest LCG PRNG for generating uniform distributed random numbers  So it may not be sufficient for some applications  But the performance comparison should be fair because all implementations uses the same PRNG  so the benchmark mainly tests the performance of the transformation

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Use std  tr1  normal distribution   The std  tr1 namespace is not a part of boost  It s the namespace that contains the library additions from the C   Technical Report 1 and is available in up to date Microsoft compilers and gcc  independently of boost

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You can use the GSL  Some complete examples are given to demonstrate how to use it

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The comp lang c FAQ list shares three different ways to easily generate random numbers with a Gaussian distribution   You may take a look of it  http   c-faq com lib gaussian html

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There are many methods to generate Gaussian-distributed numbers from a regular RNG   The Box-Muller transform is commonly used   It correctly produces values with a normal distribution   The math is easy   You generate two  uniform  random numbers  and by applying an formula to them  you get two normally distributed random numbers   Return one  and save the other for the next request for a random number

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Computer is deterministic device  There is no randomness in calculation  Moreover arithmetic device in CPU can evaluate summ over some finite set of integer numbers  performing evaluation in finite field  and finite set of real rational numbers  And also performed bitwise operations  Math take a deal with more great sets like  0 0  1 0  with infinite number of points   You can listen some wire inside of computer with some controller  but would it have uniform distributions  I don t know  But if assumed that it s signal is the the result of accumulate values huge amount of independent random variables then you will receive approximately normal distributed random variable  It was proved in Probability Theory   There is exist algorithms called - pseudo random generator  As I feeled the purpose of pseudo random generator is to emulate randomness  And the criteria of goodnes is  - the empirical distribution is converged  in some sense - pointwise  uniform  L2  to theoretical - values that you receive from random generator are seemed to be idependent  Of course it s not true from  real point of view   but we assume it s true   One of the popular method - you can summ 12 i r v with uniform distributions    But to be honest during derivation Central Limit Theorem with helping of Fourier Transform  Taylor Series  it is neededed to have n-  inf assumptions couple times  So for example theoreticaly - Personally I don t undersand how people perform summ of 12 i r v  with uniform distribution   I had probility theory in university  And particulary for me it is just a math question  In university I saw the following model     double generateUniform double a  double b      return uniformGen generateReal a  b      double generateRelei double sigma      return sigma   sqrt -2   log 1 0 - uniformGen generateReal 0 0  1 0 -kEps       double generateNorm double m  double sigma      double y2   generateUniform 0 0  2   kPi     double y1   generateRelei 1 0     double x1   y1   cos y2     return sigma x1   m        Such way how todo it was just an example  I guess it exist another ways to implement it    Provement that it is correct can be found in this book  Moscow  BMSTU  2004  XVI Probability Theory  Example 6 12  p 246-247  of Krishchenko Alexander Petrovich ISBN 5-7038-2485-0  Unfortunately I don t know about existence of translation of this book into English

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A quick and easy method is just to sum a number of evenly distributed random numbers and take their average  See the Central Limit Theorem for a full explanation of why this works

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Box-Muller implementation    include  lt cstdlib gt   include  lt cmath gt   include  lt ctime gt   include  lt iostream gt  using namespace std      return a uniformly distributed random number double RandomGenerator       return    double  rand      1       double  RAND MAX    1           return a normally distributed random number double normalRandom       double y1 RandomGenerator      double y2 RandomGenerator      return cos 2 3 14 y2  sqrt -2  log y1       int main    double sigma   82   double Mi   40     for int i 0 i lt 100 i     double x   normalRandom   sigma Mi      cout  lt  lt    x      lt  lt  x  lt  lt  endl        return 0

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I ve followed the definition of the PDF given in http   www mathworks com help stats normal-distribution html and came up with this   const double DBL EPS COMP   1 - DBL EPSILON     DBL EPSILON is defined in  lt limits h gt   inline double RandU         return DBL EPSILON     double  rand   RAND MAX     inline double RandN2 double mu  double sigma        return mu    rand   2   -1 0   1 0  sigma pow -log DBL EPS COMP RandU     0 5     inline double RandN         return RandN2 0  1 0       It is maybe not the best approach  but it s quite simple

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If you re using C  11  you can use std  normal distribution    include  lt random gt   std  default random engine generator  std  normal distribution lt double gt  distribution   mean   0 0    stddev   1 0    double randomNumber   distribution generator     There are many other distributions you can use to transform the output of the random number engine

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This is how you generate the samples on a modern C   compiler    include  lt random gt      std  mt19937 generator  double mean   0 0  double stddev    1 0  std  normal distribution lt double gt  normal mean  stddev   cerr  lt  lt   Normal     lt  lt  normal generator   lt  lt  endl

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Monte Carlo method The most intuitive way to do this would be to use a monte carlo method  Take a suitable range -X   X  Larger values of X will result in a more accurate normal distribution  but takes longer to converge  a  Choose a random number z between -X to X  b  Keep with a probability of N z  mean  variance  where N is the gaussian distribution  Drop otherwise and go back to step  a

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There exists various algorithms for the inverse cumulative normal distribution  The most popular in quantitative finance are tested on http   chasethedevil github io post monte-carlo-inverse-cumulative-normal-distribution  In my opinion  there is not much incentive to use something else than algorithm AS241 from Wichura  it is machine precision  reliable and fast  Bottlenecks are rarely in the Gaussian random number generation  The top answer here advocates for Box-M  ller  you should be aware that it has known deficiencies  I quote https   www sciencedirect com science article pii S0895717710005935   in the literature  Box   Muller is sometimes regarded as slightly inferior  mainly for two reasons  First  if one applies the Box   Muller method to numbers from a bad linear congruential generator  the transformed numbers provide an extremely poor coverage of the space  Plots of transformed numbers with spiraling tails can be found in many books  most notably in the classic book of Ripley  who was probably the first to make this observation quot

[c++] Generate random numbers following a normal distribution in C/C++

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