[从头学数学] 第196节 随机变量及其分布

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了[九转金丹]之第五转的修炼。
这次要研究的是[随机变量及其分布]。

正剧开始：

星历2016年04月26日 10:24:43, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究[随机变量及其分布]。

<span style="font-size:18px;">[0.9039684513598869, 0.09214765049540131, 0.003800386807600563, 8.247702803880503e-05, 1.026771944588782e-06, 7.505338072121047e-09, 3.1716269743581164e-11, 7.344917005316235e-14, 8.363392339049356e-17, 3.7584057247722085e-20, 3.796369418961827e-24]

#排列公式
def arrangement(n, m):
    if n < m:
        return arrangement(m, n);
    else:
        factorial = 1;
        for i in range(n, n-m, -1):
            factorial*=i;
        return factorial;

#组合公式
def combination(n, m):
    if (n < m):
        return combination(m, n);
    else:
        return arrangement(n, m)/arrangement(m,m);

#超几何分布  
def tmp():
    #假设1000件产品中有10件次品，也就是次品率1%
    #那么取出n件，其中有X件次品的概率矩阵是什么呢？
    N = 1000;
    M = 10;

    #我们取10件来查看
    n = 10;

    #X件次品, X<-[0, m], m = min{M, n}，这里m为10
    coefficient = [];

    for k in range(n+1):
        P_x_k = combination(M, k)*combination(N-M, n-k)/combination(N, n);
        coefficient.append(P_x_k);

    print(coefficient);</span>

如果现在有1000件产品在这里，其中有10件不合格，

抽查10件能找到多少次品呢？

产品质量更差一点，现在1000件里有100件不合格，

看抽查情况：

产品中有一半不合格，

抽查结果：

产品有九层不合格：

<span style="font-size:18px;">	if (1) {  
        var r = 20;        
        config.setSector(10,5,9,1);          
   
            
        //坐标轴设定    
        var scaleX = 2*r, scaleY = 2*r;      
        var spaceX = 1, spaceY = 0.125;       
        var xS = 0, xE = 10;      
        var yS = 0, yE = 1;      
    
                
        var transform = new Transform();        
        //存放函数图像上的点    
        var a = [], b = [], c = [], d = [];      
              
        //需要显示的函数说明    
        var f1 = '超几何分布N, M, n, k = 1000, 10, 10, 0~10', f2 = '10x^4-6x', f3 = '', f4 = '';    
		
		var Y = [0.9039684513598869, 0.09214765049540131, 0.003800386807600563, 8.247702803880503e-05, 1.026771944588782e-06, 7.505338072121047e-09, 3.1716269743581164e-11, 7.344917005316235e-14, 8.363392339049356e-17, 3.7584057247722085e-20, 3.796369418961827e-24];
		
		
        //函数描点    
        for (var x = xS; x <= xE; x++) {      
                a.push([x, Y[x]]);   
        }      
		              
        //存放临时数组    
        var tmp = [];    
              
        //显示变换    
        if (a.length > 0) {    
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);     
			var dx = Math.min(Math.abs(a[1][0]-a[0][0])/2, 40);
			var size = a.length;
			var y = 0;
			
			plot.setFillStyle('#FF88CC');
			for (var i = 0; i < size; i++) {
				y = a[i][1];
				shape.fillRect(a[i][0], -y/2, dx, Math.abs(y));
			}
			
            //函数1    
            tmp = [].concat(a);        
            shape.pointDraw(tmp, 'red');        
            tmp = [].concat(a);        
            shape.multiLineDraw(tmp, 'pink');      
                  
            plot.setFillStyle('red');    
            plot.fillText(f1, 30, -300, 200);      
        }  
		
		
		config.graphPaper2D(0, 0, r);        
        config.axis2D(0, 0,320, 1.6);       
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');        
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');    

	}
	</span>

<span style="font-size:18px;">[0.10879541914024672, 0.339985684813271, 0.35998484274346343, 0.1599932634415393, 0.029472443265546714, 0.0017683465959328027]

#超几何分布
#例3 联欢会
def tmp():
    N = 30;
    M = 10;

    #我们取5件来查看
    n = 5;

    #X个目标, X<-[0, m], m = min{M, n}，这里m为5
    coefficient = [];

    for k in range(n+1):
        P_x_k = combination(M, k)*combination(N-M, n-k)/combination(N, n);
        coefficient.append(P_x_k);

    print(coefficient);</span>

<span style="font-size:18px;">	if (1) {  
        var r = 20;        
        config.setSector(10,5,9,1);          
   
            
        //坐标轴设定    
        var scaleX = 2*r, scaleY = 2*r;      
        var spaceX = 0.5, spaceY = 0.125;       
        var xS = 0, xE = 5;      
        var yS = 0, yE = 1;      
    
                
        var transform = new Transform();        
        //存放函数图像上的点    
        var a = [], b = [], c = [], d = [];      
              
        //需要显示的函数说明    
        var f1 = '超几何分布N, M, n, k = 30, 10, 5, 0~5', f2 = '10x^4-6x', f3 = '', f4 = '';    
		
		var Y = [0.10879541914024672, 0.339985684813271, 0.35998484274346343, 0.1599932634415393, 0.029472443265546714, 0.0017683465959328027];
		
		
        //函数描点    
        for (var x = xS; x <= xE; x++) {      
                a.push([x, Y[x]]);   
        }      
		              
        //存放临时数组    
        var tmp = [];    
              
        //显示变换    
        if (a.length > 0) {    
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);     
			var dx = Math.min(Math.abs(a[1][0]-a[0][0])/2, 40);
			var size = a.length;
			var x = y = measure = 0;
			var s = '';
			
			plot.setFillStyle('#FF88CC');
			for (var i = 0; i < size; i++) {
				x = a[i][0];
				y = a[i][1];
				
				s = Y[i].toFixed(3);
				measure = plot.measureText(s);
				shape.fillRect(x, -y/2, dx, Math.abs(y));
				plot.fillText(s, x-measure/2,  -y-20, measure);
			}
			
			
			
            //函数1    
            tmp = [].concat(a);        
            shape.pointDraw(tmp, 'red');        
            tmp = [].concat(a);        
            shape.multiLineDraw(tmp, 'pink');      
                  
            plot.setFillStyle('red');    
            plot.fillText(f1, 30, -300, 200);      
        }  
		
		
		config.graphPaper2D(0, 0, r);        
        config.axis2D(0, 0,320, 1.6);       
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');        
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');    

	}</span>

<span style="font-size:18px;">[0.18697170950491634, 0.3983310332930826, 0.298748274969812, 0.09958275832327065, 0.015320424357426254, 0.001021361623828417, 2.4318133900676595e-05, 1.1979376305752016e-07]

#超几何分布
#题2 彩票
def tmp():
    N = 36;
    M = 7;

    #选择集容量
    n = 7;

    #X个目标, X<-[0, m], m = min{M, n}，这里m为7
    coefficient = [];

    for k in range(n+1):
        P_x_k = combination(M, k)*combination(N-M, n-k)/combination(N, n);
        coefficient.append(P_x_k);

    print(coefficient);</span>

彩票中奖率很低的。

<span style="font-size:18px;">[0.3486784401000001, 0.38742048900000015, 0.1937102445000001, 0.057395628000000025, 0.011160261000000003, 0.0014880348000000005, 0.00013778100000000007, 8.748000000000005e-06, 3.6450000000000023e-07, 9.000000000000004e-09, 1.0000000000000006e-10]

#二项分布
def tmp():
    #成功概率p
    p = 0.1;
    a = [1-p, p];
    #尝试次数/机会
    N = 10;  
  
    coefficient = [];  
  
    for i in range(N+1):  
        coefficient.append(a[0]**(N-i)*a[1]**i*combination(N, i));  
  
    print(coefficient);        </span>

<span style="font-size:18px;">	if (1) {  
        var r = 20;        
        config.setSector(10,5,9,1);          
   
            
        //坐标轴设定    
        var scaleX = 2*r, scaleY = 2*r;      
        var spaceX = 1, spaceY = 0.125;       
        var xS = 0, xE = 10;      
        var yS = 0, yE = 1;      
    
                
        var transform = new Transform();        
        //存放函数图像上的点    
        var a = [], b = [], c = [], d = [];      
              
        //需要显示的函数说明    
        var f1 = '二项分布x~B(10, 0.1)', f2 = '', f3 = '', f4 = '';    
		
		var Y = [0.3486784401000001, 0.38742048900000015, 0.1937102445000001, 0.057395628000000025, 0.011160261000000003, 0.0014880348000000005, 0.00013778100000000007, 8.748000000000005e-06, 3.6450000000000023e-07, 9.000000000000004e-09, 1.0000000000000006e-10];
		
		
        //函数描点    
        for (var x = xS; x <= xE; x++) {      
                a.push([x, Y[x]]);   
        }      
		              
        //存放临时数组    
        var tmp = [];    
              
        //显示变换    
        if (a.length > 0) {    
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);     
			var dx = Math.min(Math.abs(a[1][0]-a[0][0])/2, 40);
			var size = a.length;
			var x = y = measure = 0;
			var s = '';
			
			plot.setFillStyle('#FF88CC');
			for (var i = 0; i < size; i++) {
				x = a[i][0];
				y = a[i][1];
				
				s = Y[i].toFixed(3);
				measure = plot.measureText(s);
				shape.fillRect(x, -y/2, dx, Math.abs(y));
				plot.fillText(s, x-measure/2,  -y-20, measure);
			}
			
			
			
            //函数1    
            tmp = [].concat(a);        
            shape.pointDraw(tmp, 'red');        
            tmp = [].concat(a);        
            shape.multiLineDraw(tmp, 'pink');      
                  
            plot.setFillStyle('red');    
            plot.fillText(f1, 30, -300, 200);      
        }  
		
		
		config.graphPaper2D(0, 0, r);        
        config.axis2D(0, 0,320, 1.6);       
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');        
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');    

	}</span>

<span style="font-size:18px;">[1.0239999999999978e-07, 4.095999999999993e-06, 7.372799999999987e-05, 0.000786431999999999, 0.005505023999999994, 0.026424115199999976, 0.08808038399999996, 0.20132659199999994, 0.301989888, 0.26843545600000007, 0.10737418240000006]

#二项分布
#例4 射击
def tmp():
    #成功概率p
    p = 0.8;
    a = [1-p, p];
    #尝试次数/机会
    N = 10;  
  
    coefficient = [];  
  
    for i in range(N+1):  
        coefficient.append(a[0]**(N-i)*a[1]**i*combination(N, i));  
  
    print(coefficient);     
	</span>

<span style="font-size:18px;">3.5000000000000004
2.916666666666667
1.7078251276599332

	if (1) {
		var analyze = new DataAnalyze();
		
		var array = [];
		
		for (var x = 1; x <=6; x++) {
			array.push([x, 1/6]);
		}
		
		var E_x = analyze.weightedAverage(array);
		var D_x = analyze.variance(array);
		var sqrtD_x = analyze.RMS(array);
		
		document.write(E_x+'<br/>');
		document.write(D_x+'<br/>');
		document.write(sqrtD_x+'<br/>');
	
	}
	</span>

<span style="font-size:18px;">1400.0000000000002
40000.000000000015
200.00000000000003
-------------
1400.0000000000002
160000.00000000003
400.00000000000006

	if (1) {
		var analyze = new DataAnalyze();
		
		var A = [[1200,0.4],[1400,0.3],[1600,0.2],[1800,0.1]];
		var B = [[1000,0.4],[1400,0.3],[1800,0.2],[2200,0.1]];
		

		
		var E_x = analyze.weightedAverage(A);
		var D_x = analyze.variance(A);
		var sqrtD_x = analyze.RMS(A);
		
		document.write(E_x+'<br/>');
		document.write(D_x+'<br/>');
		document.write(sqrtD_x+'<br/>');
		
		document.write('-------------'+'<br/>');
		
		E_x = analyze.weightedAverage(B);
		D_x = analyze.variance(B);
		sqrtD_x = analyze.RMS(B);
		
		document.write(E_x+'<br/>');
		document.write(D_x+'<br/>');
		document.write(sqrtD_x+'<br/>');
		
		
	
	}</span>

<span style="font-size:18px;">/**
* @usage   数据的集中分析
* @author  mw
* @date    2016年03月22日  星期二  08:24:39 
* @param   原始数据数组
* @return
*
*/
function DataAnalyze() {
	//小数位数超过精度的数会四舍五入到指定精确位数
	//但如果小数位数小于要求精度，不会进行处理
	//整数部分加上精度位数不超过15个数字可以达到要求
	this.round = function(number, precision) {
		var scale = Math.pow(10, precision);
		
		return Math.round(number*scale)/scale;
	}
	
	
	//平均数， 数组为[num1, num2, ... ]
	this.average = function(array) {
		if (array[0].length) 
			return Number.NaN;
			
		if (array == []) {
			return 0;
		}
		else {
			var len = array.length;
			var sum = 0;
			for (var i = 0; i < len; i++) {
				sum += array[i];
			}
			return sum/len;
		}
	}
	
	//加权平均数， 数组为[[num1, weight1], [num2,weight2],  ... ]
	this.weightedAverage = function(array) {
		if (!array[0].length) 
			return Number.NaN;
			
		if (array == []) {
			return 0;
		}
		else {
			var len = array.length;
			var sum = 0;
			var totalWeight = 0;
			for (var i = 0; i < len; i++) {
				sum += array[i][0]*array[i][1];
				totalWeight += array[i][1];
			}
			return sum/totalWeight;
		}
	}
	
	//中位数, 数组为[num1, num2, ... ]
	this.median = function(array) {
		if (array == []) {
			return 0;
		}
		else {
			array.sort(function(a, b) {
				return a - b;
			});
			
			var len = array.length;
			
			if (len%2 == 1) {
				//样本总数为奇数个
				return array[(len-1)/2];
			}
			else {
				//样本总数为偶数个
				return (array[len/2-1]+array[len/2])/2;
			}
		}
	}
	
	//极差, 数组为[num1, num2, ... ]
	this.extremeDifference = function(array) {
		if (array == []) {
			return 0;
		}
		else {
			array.sort(function(a, b) {
				return a - b;
			});
			
			var len = array.length;
			
			return Math.abs(array[0] - array[len-1]);
		}
	}
	
	//众数, 数组为[[num1, weight1], [num2,weight2],  ... ]
	this.mode = function(array) {
		if (array == []) {
			return 0;
		}
		else {
			var newArray = [];
			newArray = this.normalToWeightedArray(array);

			
			//根据权重进行排序
			newArray.sort(function(a, b) {
				return -(a[1] - b[1]);
			});
			
			//返回权重最大的数
			return newArray[0][0];
		}
	}
	
	//方差, 数组为[num1, num2, ... ]
	this.variance = function(array) {
		if (array == []) {
			return 0;
		}
		else {
			var newArray = new Array();
			newArray = array;
			
			if (!newArray[0].length) {
				//一维数组，array[0].length无定义
				var average = this.average(newArray);
				var len = newArray.length;
				
				var sum = 0;
				
				for (var i = 0; i < len; i++) {
					sum += Math.pow((newArray[i]-average), 2);
				}
				
				return sum/len;
			}
			else if (newArray[0].length > 1) {
				var average = this.weightedAverage(newArray);
				var len = newArray.length;
				
				var sum = 0;
				var count = 0;
				
				for (var i = 0; i < len; i++) {
					sum += Math.pow((newArray[i][0]-average), 2)*newArray[i][1];
					count += newArray[i][1];
				}
				
				return sum/count;
			}
		}
	}
	
	//标准差(root-mean-square deviation) , 数组为[num1, num2, ... ]	
	this.RMS = function(array) {
		return Math.sqrt(this.variance(array));
	}
	
	//平均差 , 数组为[num1, num2, ... ]
	this.meanDifference = function(array) {
		if (array == []) {
			return 0;
		}
		else {
			var average = this.average(array);
			var len = array.length;
			
			var sum = 0;
			
			for (var i = 0; i < len; i++) {
				sum += Math.abs(array[i]-average);
			}
			
			return sum/len;
		}
	
	}
	
	
	//把[[num1, weight1], [num2,weight2],  ... ]形式的数组转化成
	//[num1, num2, ... ]形式的数组
	this.weightedToNormalArray = function(array) {
		if (array == []) {
			return [];
		}
		else {
			var len = array.length;
			if (array[0].length < 2) {
				return array;
			}
			else {
				var retArray = new Array();
				var times = 0;
				
				for (var i = 0; i < len; i++) {
					//权重转为重复次数
					times = array[i][1];
					
					for (var j = 0; j < times; j++) {
						retArray.push(array[i][0]);
					}
				}
				return retArray;
			}			
			
		}
	}
					
	//把[num1, num2, ... ]形式的数组转化成
	//[[num1, weight1], [num2,weight2],  ... ]形式的数组
	this.normalToWeightedArray = function(array) {
		if (array == []) {
			return [];
		}
		else {
			var len = array.length;
			var len2 = 0;
			
			if (array[0].length > 1) {
				return array;
			}
			else {
				var retArray = new Array();
				retArray.push([array[0], 1]);
				
				for (var i = 1; i < len; i++) {
					len2 = retArray.length;					
					
					for (var j = 0; j < len2; j++) {
						if (Math.abs(array[i]-retArray[j][0]) < 0.0001) {
							retArray[j][1]+=1;
							break;
						}
						
						if (j >= len2 - 1) {
							retArray.push([array[i], 1]);
						}
						
					}
				}
				return retArray;
			}
			
		}
	}
	
	//样本数
	this.sampleSize = function(array) {
		if (!array[0].length) {
			return array.length;
		}
		else {
			var size = array.length;
			var sum = 0;
			for (var i = 0; i < size; i++) {
				sum += array[i][1];
			}
			
			return sum;
		}
	}
	
	//加权数组计算频率
	this.rateCalc = function(array) {
		if (!array[0].length) {
			array = this.normalToWeightedArray(array);
			
			return this.rateCalc(array);
		}
		else {
			var size = array.length;
			var sum = this.sampleSize(array);
			var retArray = new Array();
			for (var i = 0; i < size; i++) {
				retArray.push([array[i][0], array[i][1]/sum]);
			}
			
			return retArray;
		}
	}
	
		
}</span>

<span style="font-size:18px;">	if (1) {  
        var r = 20;        
        config.setSector(10,5,9,1);          
   
            
        //坐标轴设定    
        var scaleX = 2*r, scaleY = 1.5*r;      
        var spaceX = 1, spaceY = 0.05;       
        var xS = 0, xE = 10;      
        var yS = 0, yE = 1;      
    
                
        var transform = new Transform();        
        //存放函数图像上的点    
        var a = [], b = [], c = [], d = [];      
              
        //需要显示的函数说明    
        var f1 = '正态分布x~1/((2pi)^0.5σ)exp^(-(x-μ)^2/2σ^2)', f2 = '', f3 = '', f4 = '';    
		var Y = [];
		var tmpY = 0;
		
		
        //函数描点    
        for (var x = xS; x <= xE; x+=0.2) {     
			 tmpY = funTask(x);
             a.push([x, tmpY]);   
			 Y.push(tmpY);

        }      
		              
        //存放临时数组    
        var tmp = [];    
              
        //显示变换    
        if (a.length > 0) {    
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);     
			var dx = Math.min(Math.abs(a[1][0]-a[0][0])/2, scaleX/2);
			var size = a.length;
			var x = y = measure = 0;
			var s = '';
			
			plot.setFillStyle('#FF88CC');
			for (var i = 0; i < size; i++) {
				x = a[i][0];
				y = a[i][1];
				
				shape.fillRect(x, -y/2, dx, Math.abs(y));
				
				
				/*s = Y[i].toFixed(3);
				measure = plot.measureText(s);				
				plot.fillText(s, x-measure/2,  -y-20, measure);*/
				
				
			}
			
			
			
            //函数1    
            tmp = [].concat(a);        
            shape.pointDraw(tmp, 'red');        
            tmp = [].concat(a);        
            shape.multiLineDraw(tmp, 'pink');      
                  
            plot.setFillStyle('red');    
            plot.fillText(f1, 30, -300, 200);      
        }  
		
		
		config.graphPaper2D(0, 0, r);        
        config.axis2D(0, 0,320, 1.6);       
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');        
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');    

	}

function funTask(x) {
	var mju = 5;
	var sigma = 1;
	
	return 1/Math.sqrt(2*Math.PI)/sigma*Math.pow(Math.E, -(x-mju)*(x-mju)/2/sigma/sigma);
}
</span>

本节到此结束，欲知后事如何，请看下回分解。