连续型随机变量
- 无穷多取值
- 概率分布 \(\Longrightarrow\) 概率密度函数 PDF
\[\int f(x)dx = 1\]
- 累积分布函数 CDF
\[f(x) = F'(x)\]
\[F(x) = P(X\leq x) = \int_{-\infty}^x f(t)dt\]
例
- 质子中不同类别的夸克和胶子（部分子）携带动量比例的概率密度
连续均匀分布（Uniform）¶
\[f(x) = \begin{cases} \frac{1}{b-a}, & a\leq x\leq b \\ 0, & \text{otherwise} \end{cases} \]
\[E(X) = \int_{a}^b x\frac{1}{b-a}dx = \frac{a+b}{2}\]
\[\text{var}(X) = \int_{a}^b x^2\frac{1}{b-a}dx - \left(\frac{a+b}{2}\right)^2 = \frac{(b-a)^2}{12}\]
Python 实现¶
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import scipy.stats as st
x = st.uniform(2,5).rvs(size=1000) # 2-7
st.uniform(2,5) 表示 \([2,7)\)
- 绘制 PDF
x = np.arange(2,7,0.01)
sns.histplot(x, bins=x, stat='density')
f = st.uniform(2,5).pdf(x)
plt.plot(x, f, 'r')
指数分布（Exponential）¶
\[f(x) = \begin{cases} \lambda e^{-\lambda x}, & x\geq 0 \\ 0, & \text{otherwise} \end{cases} \]
- 描述某事件发生所需要的时间
- 单位时间事件发生的次数平均值为 \(\lambda\)，时长 \(t\) 之内发生的总次数为 \(\lambda t\). 实际观察到的次数符合泊松分布： \(\(g(n) = \frac{e^{-\lambda t}(\lambda t)^n}{n!}\)\)
设 \(0 \sim t\) 内事件未发生，其概率为 \(g(0) = e^{-\lambda t}\).
\(t \sim t+dt\) 内事件发生一次的概率为
正态分布（Normal）/ 高斯分布（Gaussian）¶
\[f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]
- 期望值 \(\mu\)
- 标准差 \(\sigma\)
Python 实现¶
二维正态分布
\[f(x,y) = \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}\text{exp}\left[{-\frac{1}{2(1-\rho^2)}\left(\frac{(x-\mu_1)^2}{\sigma_1^2} + \frac{(y-\mu_2)^2}{\sigma_2^2} - \frac{2\rho(x-\mu_1)(y-\mu_2)}{\sigma_1\sigma_2}\right)}\right]\]
协方差矩阵
\[\Sigma = \begin{bmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{bmatrix}\]
Python 实现
条件分布¶
rv = multivariate_normal([xmean,ymean], cov)
fig = plt.figure()
ax = fig.add_subplot(111)
x1d = np.arange(xmin,xmax,.1)
y1d = np.arange(-2,-2,.06)
for yval in [-2,-1,0]:
z=[]
for i in x1d:
z.append(rv.pdf([i,yval]))
con=ax.plot(x1d,z/sum(z),label="y={:.1f}".format(yval))
plt.legend()
边缘分布¶
from scipy.stats.contingency import margins
x1d = np.arange(xmin, xmax, 0.1)
y1d = np.arange(ymin, ymax, 0.1)
xmargin, ymargin = margins(rv.pdf(pos))
ax3.plot(x1d,xmargin/sum(xmargin),y1d,ymargin.T/sum(ymargin.T))
协方差和相关系数¶
- 协方差 \(\text{cov}(X,Y) = E[(X-\mu_X)(Y-\mu_Y)] = E(XY) - E(X)E(Y)\)
- 若 \(X,Y\) 独立，则 \(\text{cov}(X,Y) = 0\)
- 相关系数 \(\rho = \frac{\text{cov}(X,Y)}{\sigma_X\sigma_Y}\)，也可记作 \(\text{corr}(X,Y)\)
- 协方差矩阵 \(\Sigma = \begin{bmatrix} \text{var}(X) & \text{cov}(X,Y) \\ \text{cov}(X,Y) & \text{var}(Y) \end{bmatrix}\)
- 二维正态分布的 \(\text{Cov}(X,Y) = \rho\sigma_X\sigma_Y\), \(\text{Corr}(X,Y) = \rho\)