Standard Uniform Distribution

By Hubert Ronald / Leave a response / May 28, 2018

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In statistics, the antithetic variates method is a variance reduction technique used in Monte Carlo methods. This last is an interesting property.

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable From Wikipedia

Where its cumulative distribution function is:

\begin{equation} F(x) = \begin{cases} 0, & \text{for}\ \ x < a \\ \frac{x-a}{b-a}, & \text{for}\ \ a \leq x \leq b \\ 1, & \text{for}\ \ x > b \end{cases} \end{equation}

Its inverse is:

\( F^{-1}(p) =a+p(b-a) \text{ for } 0<\text{p}<1 \)

if \(a=0\), \(b=1\) and \(p=u\) Standard Uniform Distribution Inverse is:

\( F^{-1}(u) = u \text{ for } 0<\text{u}<1 \)

Note that if \(u=1-u^*\) new Standard Uniform Distribution Inverse is:

\( F^{-1}(u^*) = u^* \text{ for } 0< u^* <1 \)

And the above is known as antithetic variates method. Next, its implementation in C++ is shown:

#include <iostream>
#include <cstdlib>
#include <ctime>

using namespace std;
//-- Standard Uniform Distribution  --
inline double unif(){return ((double)rand()/(double)RAND_MAX);} 

int main()
{
    int run = 50;   //-- Value according to your needs  --
    srand(clock());

    //-- note than expression 1-unif() is going to equal be to unif()  --
    for (int i = 0; i < run; ++i){cout << i<<"   " << 1-unif() << endl;};
    system("pause");
    return 0;
}

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