Notes on probability theory: random numbers, probability distribution (normal distribution), central limit theorem (theorem of large numbers)

1. Probability uniform distribution (C language)

#include <stdio.h>
#include <stdlib.h>
#include <time.h>

int main( int argc, char **argv)
{
	int i=0, v, mod, n;
	/* The srand function is the initialization function of the random number generator */
	srand((unsigned) time(NULL));//This line can be deleted

	mod = 10;//range of random
	n = 100000;//number of random
	
	int *count = malloc(sizeof(int)*mod);
	i=0;
	do{
		count[i++] = 0;
	}while(i<mod);
	
	i=0;
	do{
		i++;
		v = rand() % mod;
		printf("%2d ", v);
		if(i%10 == 0)
			printf("\n");
		count[v] ++;
		
	}while(i<n);
	
	i=0;
	do{
		printf("count[%d] = %d\n",i, count[i]);
		i++;
	}while(i<mod);
	
	return 0;
}

result:

count[0] = 9973
count[1] = 10146
count[2] = 9909
count[3] = 9837
count[4] = 10079
count[5] = 10111
count[6] = 9999
count[7] = 10002
count[8] = 9977
count[9] = 9967

2. Normal distribution formula (C language)

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <math.h>

#ifndef pi
#define pi 3.141592653
#endif

int main( int argc, char **argv)
{
	int mu = 5, cigma = 1;

	int n, ix;
	float dn, x;
	
	n = 100;
	dn = 0.1;
	
	float *N = malloc(sizeof(float)*n);
	
	for (ix = 0, x = 0; ix <n; ix ++, x+= dn)
	{
		printf("%d, %f\n",ix, x);
		N[ix] = sqrt(1.0/(2*pi*cigma*cigma))*exp( -1.0/(2.0*cigma*cigma)*(x - mu)*(x - mu) );
	}
	
	FILE *fp = fopen("normal.txt","w");
	for (ix = 0; ix <n; ix ++)
		fprintf(fp,"%f\n",N[ix]);
	fclose(fp);
	return 0;
}

The result is pictured:

3. Central Limit Theorem (Theorem of Large Numbers)

The definition of the theorem is that the sum of many random variables approximately obeys the normal distribution, and the code is given and the parameters are changed to plot:

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <math.h>

#ifndef pi
#define pi 3.141592653
#endif

float Normal_0(float mu, float cigma, float x)
/* normal distribution with fmulation */
{
	return sqrt(1.0/(2*pi*cigma*cigma))
				*exp( -1.0/(2.0*cigma*cigma)*(x - mu)*(x - mu) );
}

#define NSUM 250
int Normal_1()
/*Theorem of large numbers
 or Central Limit Theorem
*/
{
	srand((unsigned) time(NULL));
    float x = 0;
    int i;
    for(i = 0; i < NSUM; i++)
    {
        x += rand() %10;
		//printf("%f\n",x);
    }printf("%f\n",x);
    return (int)(x/NSUM);
}

int main( int argc, char **argv)
{
	srand((unsigned) time(NULL));
	
	int i, j;
	int n = 10000, RAN_MAX = 1000, SUM, mod = 100;
	
	int *count = malloc(4*mod);
	for(i=0;i<mod;i++)
		count[i] = 0;

	FILE *fp = fopen("tmp.txt","wb");
	int min = 9999999, max = -9999999;
	for(i=0;i<n;i++)
	{
		SUM = 0;
		for(j=0;j<RAN_MAX;j++)
		{
			SUM += rand() % mod;
		}
		fprintf(fp,"%d\n",SUM);
	}
	fclose(fp);
	fp = fopen("tmp.txt","r");
	for(i=0;i<n;i++)
	{
		fscanf(fp,"%d\n",&SUM);
		if(SUM < min)
			min = SUM;
		if(SUM > max)
			max = SUM;
	}
	printf("min = %d, max = %d\n",min,max);
	rewind(fp);
	for(i=0;i<n;i++)
	{
		fscanf(fp,"%d\n",&SUM);
		SUM = (int)((float)(SUM-min)/(max-min)*mod);
		count[SUM] ++;
	}
	for(i=0;i<mod;i++)
	{
		printf("%d\n",count[i]);
	}
	
	return 0;
}

A temporary file "tmp.txt" will be generated in the middle, which stores the sum of random variables. We "normalize" these parameters and draw the following figure: