beta（贝塔）分布、用python从beta分布抽样并画出直方图

系统管理员 2024-04-23 12:44 65阅读 0赞

## beta分布的表达式 ##

beta分布的概率密度函数为：

![f(x;\\alpha ,\\beta )=\\frac\{1\}\{B(\\alpha ,\\beta )\}x^\{\\alpha -1\}(1-x)^\{\\beta -1\}][f_x_alpha _beta _frac_1_B_alpha _beta _x_alpha -1_1-x_beta -1]

其中归一化![B(\\alpha ,\\beta )][B_alpha _beta]是beta函数：

![B(\\alpha ,\\beta )=\\int\_\{0\}^\{1\}t^\{\\alpha -1\}(1-t)^\{\\beta -1\}dt][B_alpha _beta _int_0_1_t_alpha -1_1-t_beta -1_dt]

式中![x][]定义在![(0,1)][0_1]区间，参数![\\alpha ,\\beta >0][alpha _beta _0]

## 用python从beta分布中抽样并画出直方图 ##

import numpy as np
    import matplotlib.pyplot as plt
    
    a = 0.5
    b = 0.5
    # 从贝塔分布抽样
    x = np.random.beta(a, b, size=10000)
    # 在数据集x上计算并画出直方图
    plt.hist(x, bins='auto', label=f'a={a}, b={b}')
    
    plt.legend()
    plt.show()

![e15efafb31f644ebb0c72e3af2c2b81a.png][]

改变参数![\\alpha][alpha]和![\\beta][beta]的数值，得到的不同的直方图：

![a5deef080d034aa7b6120a3b340df861.png][]

![757a813aee474513a08b6fbf0bcdc098.png][]

![88be171431e74facb5b1c5155d262dad.png][]

![1a2091ab9bdc4edfb18113de257071f5.png][]

[f_x_alpha _beta _frac_1_B_alpha _beta _x_alpha -1_1-x_beta -1]: https://latex.csdn.net/eq?f%28x%3B%5Calpha%20%2C%5Cbeta%20%29%3D%5Cfrac%7B1%7D%7BB%28%5Calpha%20%2C%5Cbeta%20%29%7Dx%5E%7B%5Calpha%20-1%7D%281-x%29%5E%7B%5Cbeta%20-1%7D
[B_alpha _beta]: https://latex.csdn.net/eq?B%28%5Calpha%20%2C%5Cbeta%20%29
[B_alpha _beta _int_0_1_t_alpha -1_1-t_beta -1_dt]: https://latex.csdn.net/eq?B%28%5Calpha%20%2C%5Cbeta%20%29%3D%5Cint_%7B0%7D%5E%7B1%7Dt%5E%7B%5Calpha%20-1%7D%281-t%29%5E%7B%5Cbeta%20-1%7Ddt
[x]: https://latex.csdn.net/eq?x
[0_1]: https://latex.csdn.net/eq?%280%2C1%29
[alpha _beta _0]: https://latex.csdn.net/eq?%5Calpha%20%2C%5Cbeta%20%3E0
[e15efafb31f644ebb0c72e3af2c2b81a.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/23/c926b06647b04c3ca25246906e5cd1e4.png
[alpha]: https://latex.csdn.net/eq?%5Calpha
[beta]: https://latex.csdn.net/eq?%5Cbeta
[a5deef080d034aa7b6120a3b340df861.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/23/5ad24d06efa54b6fab5252cb00247fc8.png
[757a813aee474513a08b6fbf0bcdc098.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/23/008d1a01812c43e6ae1fa9743820de76.png
[88be171431e74facb5b1c5155d262dad.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/23/d39a194d58864fe0b283419232d58225.png
[1a2091ab9bdc4edfb18113de257071f5.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/23/db9f72218e8c43f0b21f1cad6597952a.png