1.涉及公式

1.1 高斯分布公式

概率密度函数

1.2 二项分布公式

换句话说，一枚公平的硬币有正面结果的概率（正面）p = 0.5。如果你掷硬币 20 次，平均值为 20 * 0.5 = 10；你会期望得到10个正面

1.3 方差

继续以硬币为例，n 是投掷硬币的次数，p 是正面朝上的概率

1.4 标准差

换句话说，标准差是方差的平方根。

1.5 概率密度函数

2.编写高斯类

import math
import matplotlib.pyplot as plt

class Gaussian():
    """ 高斯分布类，用于计算和可视化高斯分布.

    Attributes:
        均值（float），表示分布的均值。stdev（float）表示分布数据的标准偏差。_ list（float列表）：从数据文件中提取的浮点列表

    """
    def __init__(self, mu = 0, sigma = 1):

        self.mean = mu
        self.stdev = sigma
        self.data = []


    def calculate_mean(self):

        """函数计算数据集的平均值.

        Args: 
            None

        Returns: 
            float: mean of the data set

        """

        avg = 1.0 * sum(self.data) / len(self.data)

        self.mean = avg

        return self.mean



    def calculate_stdev(self, sample=True):

        """函数计算数据集的标准偏差.

        Args: 
            sample (bool): 数据是代表样本还是总体

        Returns: 
            float: 数据集的标准偏差

        """

        if sample:
            n = len(self.data) - 1
        else:
            n = len(self.data)

        mean = self.mean

        sigma = 0

        for d in self.data:
            sigma += (d - mean) ** 2

        sigma = math.sqrt(sigma / n)

        self.stdev = sigma

        return self.stdev


    def read_data_file(self, file_name, sample=True):

        """函数从txt文件读入数据。txt文件应该具有每行一个数字（浮动）。这些数字存储在数据属性中。读取文件后，计算平均值和标准偏差

        Args:
            file_name (string): name of a file to read from

        Returns:
            None

        """

        with open(file_name) as file:
            data_list = []
            line = file.readline()
            while line:
                data_list.append(int(line))
                line = file.readline()
        file.close()

        self.data = data_list
        self.mean = self.calculate_mean()
        self.stdev = self.calculate_stdev(sample)


    def plot_histogram(self):
        """函数使用matplotlib pyplot库输出实例变量数据的直方图.

        Args:
            None

        Returns:
            None
        """
        plt.hist(self.data)
        plt.title('Histogram of Data')
        plt.xlabel('data')
        plt.ylabel('count')



    def pdf(self, x):
        """高斯分布的概率密度函数计算器.

        Args:
            x (float): 计算概率密度函数的点


        Returns:
            float: 输出的概率密度函数
        """

        return (1.0 / (self.stdev * math.sqrt(2*math.pi))) * math.exp(-0.5*((x - self.mean) / self.stdev) ** 2)


    def plot_histogram_pdf(self, n_spaces = 50):

        """函数绘制数据的归一化直方图，并沿相同范围绘制概率密度函数

        Args:
            n_spaces (int): number of data points 

        Returns:
            list: x values for the pdf plot
            list: y values for the pdf plot

        """

        mu = self.mean
        sigma = self.stdev

        min_range = min(self.data)
        max_range = max(self.data)

         # 计算x值之间的间隔
        interval = 1.0 * (max_range - min_range) / n_spaces

        x = []
        y = []

        # calculate the x values to visualize
        for i in range(n_spaces):
            tmp = min_range + interval*i
            x.append(tmp)
            y.append(self.pdf(tmp))

        # make the plots
        fig, axes = plt.subplots(2,sharex=True)
        fig.subplots_adjust(hspace=.5)
        axes[0].hist(self.data, density=True)
        axes[0].set_title('Normed Histogram of Data')
        axes[0].set_ylabel('Density')

        axes[1].plot(x, y)
        axes[1].set_title('Normal Distribution for \n Sample Mean and Sample Standard Deviation')
        axes[0].set_ylabel('Density')
        plt.show()

        return x, y

3.测试高斯类

import unittest

class TestGaussianClass(unittest.TestCase):
    def setUp(self):
        self.gaussian = Gaussian(25, 2)

    def test_initialization(self): 
        self.assertEqual(self.gaussian.mean, 25, 'incorrect mean')
        self.assertEqual(self.gaussian.stdev, 2, 'incorrect standard deviation')

    def test_pdf(self):
        self.assertEqual(round(self.gaussian.pdf(25), 5), 0.19947,\
         'pdf function does not give expected result') 

    def test_meancalculation(self):
        self.gaussian.read_data_file('numbers.txt', True)
        self.assertEqual(self.gaussian.calculate_mean(),\
         sum(self.gaussian.data) / float(len(self.gaussian.data)), 'calculated mean not as expected')

    def test_stdevcalculation(self):
        self.gaussian.read_data_file('numbers.txt', True)
        self.assertEqual(round(self.gaussian.stdev, 2), 92.87, 'sample standard deviation incorrect')
        self.gaussian.read_data_file('numbers.txt', False)
        self.assertEqual(round(self.gaussian.stdev, 2), 88.55, 'population standard deviation incorrect')

tests = TestGaussianClass()

tests_loaded = unittest.TestLoader().loadTestsFromModule(tests)

unittest.TextTestRunner().run(tests_loaded)