Monte Carlo function 

1. Definition:

Monte Carlo technique is generally used to intergrate functions 
using random numbers. Here is the method:




2. Examples in C language:

//1.Integral:
//-------------
	
	#include <stdio.h>
	#include <stdlib.h>
	#include <time.h>
	#include <math.h>
	#define pi 3.1416

	int main()
	{
	float a ;
	float b ;
	int number;
	printf("\n Enter two limit numbers and a long integer : -->  ");
	scanf("%f %f %d", &a,&b,&number);

	float n = 1.0/number;
	float y;
	float random_x;
	float random_y;
	int count=0;

	int i = 0;
	for (i =1; i <= number; i++)
	{
	unsigned int seed = (unsigned int)time(NULL);
	srand (i*seed);
	random_x = a + (rand()/(RAND_MAX +1.0))*(b - a);
	random_y = a + (rand()/(RAND_MAX +1.0))*(3.0 + 1.0);
	// choose a function
	//------------------
	y = 2*random_x;
	if(y  >= random_y )
	{
	count =count + 1;
	}

	}
	printf("The rate is %d \n",count);
	printf("\n");

	float m;
	m	= count*n;
	float s = m*8;
	printf("The rate is %2.3f\n",m);
	printf("The area is %4.3f\n",s);
	return 0;
	}

	
	//2. Other function sin(x)
	//-------------------------
	#include <stdio.h>
	#include <stdlib.h>
	#include <time.h>
	#include <math.h>
	#define pi 3.1416

	int main()
	{
	float a ;
	float b ;
	int number;
	float surface;
	printf("\n Enter two limit numbers, a long integer, and the surface : -->  ");
	scanf("%f %f %d %f", &a,&b,&number,&surface);
	//In this example, the surface = pi
	float n = 1.0/number;
	float y;
	float random_x;
	float random_y;
	int count=0;

	int i = 0;
	for (i =1; i <= number; i++)
	{
	unsigned int seed = (unsigned int)time(NULL);
	srand (i*seed);
	random_x = a + (rand()/(RAND_MAX +1.0))*(b - a);
	random_y = a + (rand()/(RAND_MAX +1.0))*(0.0 + 1.0);
	y = sin(random_x);
	//chosen function : sin(x)
	//-------------------------
	if(y  >= random_y )
	{
	count = count + 1;
	}

	}
	printf("The rate is %d \n",count);
	printf("\n");

	float m;
	m	= count*n;
	float s = m*surface;
	printf("The rate is %2.3f\n",m);
	printf("The area is %4.3f\n",s);
	return 0;
	}

	
//3. Calculating the number pi:
//-------------------------------
	#include <stdio.h>
	#include <stdlib.h>
	#include <time.h>
	#include <math.h>


	int main()
	{
	int number;
	printf("\n Enter a long integer : -->  \n");
	scanf("%d" , &number);

	float n = 1.0/number;

	float yo;
	float y;
	float random_x;
	float random_y;
	int count=0;

	int i = 0;
	for (i =1; i <= number; i++)
	{
	unsigned int seed = (unsigned int)time(NULL);
	srand (i*seed);
		//generating random numbers between 0 and 1
		random_x = 0.0 + (rand()/(RAND_MAX +1.0))*(0.0 + 1.0);
		random_y = 0.0 + (rand()/(RAND_MAX +1.0))*(0.0 + 1.0);

	yo = pow(random_x,2) +  pow(random_y,2);
	y = sqrt(yo);

	if(y  < = 1)/*condition to be within the circle: the distance y must be
		lower or egal to  the radius if of the circle which is equal to 1. */
	{
	count = count + 1;
	}

	}
	printf("The hit count is %d \n", count);
	printf("\n");

	float m;
	m	= count*n;
	float surface = 1;
	// = 1 x 1 , the surface of the square
	float s = m*surface;
	printf("The rate count/number is %2.3f\n",m);
	printf("\n");
	printf("The value of pi/4 is %4.3f\n",s);
	printf("\n");
	printf("The value of pi is %4.3f\n",4*s);
	return 0;
	}
	
//4. Using Gaussain distribution to calculate a probability
// -------------------------------------------------------------
	
	#include <stdio.h>
	#include <stdlib.h>
	#include <time.h>
	#include <math.h>
	#define pi 3.1416

	int main()
	{

	float Po = 2*sqrt(pi);
	float  P = 1.0/Po;

	float a ;
	float b ;
	int number;
	printf("\n Enter two limit numbers, and a long integer : -->  ");
	scanf("%f %f %d", &a,&b,&number);

	float n = 1.0/number;
	float y;
	float random_x;
	float random_y;
	int count=0;

	int i = 0;
	for (i =1; i<= number; i++)
	{
	unsigned int seed = (unsigned int)time(NULL);
	srand (i*seed);
	random_x = a + (rand()/(RAND_MAX +1.0))*(b - a);
	random_y = 0.0 + (rand()/(RAND_MAX +1.0))*(0.0 + 1.0);
		float w = 0.5*pow(random_x,2);
	y = exp (-w);
	if(y  >= random_y )
	{
	count = count + 1;
	}

	}
	printf("The hit count is %d \n",count);
	printf("\n");

	float m;
	float surface;
	surface = 1*(b-a);
	m = count*n;
	float s = P*m*surface ;
	printf("The rate count/number is %6.6f\n",m);
	printf("The area is %4.3f\n",s);
	return 0;

	}

	/*
	Notes:
	--------
	1.
	count = ∑ f(random_x)
	(i: from 1 to number)
	the function f is the hit-count function
	P = 1/Po = 1/sqrt(2*pi) for normalisation
		f = P*count
		s P*m*(b-a) = P*count*n*(b-a)= P*count*(b-a)/number
	Thus the area is:
	s = [(b-a)/number] *  ∑ f(random_x)
	(i: from 1 to number)

	2.
	Here, the area is the probability to find an event between a and b,
	using Gauss probability distribution.
	*/