概率论笔记：随机数、概率分布（正态分布）、中心极限定理（大数定理）

1.概率均匀分布（C语言）

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

int main( int argc, char **argv)
{ 
	int i=0, v, mod, n;
	/* srand函数是随机数发生器的初始化函数 */
	srand((unsigned) time(NULL));//此行可以删除

	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;
}

结果：

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.正太分布公式（C语言）

#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;
}

结果成图：

3.中心极限定理（大数定理）

该定理的定义为很多随机变量的和近似服从正太分布，给出代码并改变参数作图：

#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;
}

中间会生成临时文件“tmp.txt”，里面保存了随机变量的和，我们对这些参数进行“归一化”，并作图得出下图：