衡水建网站的公司,一般家庭装修照片,网站建设需要上传数据库吗,公司起名字大全免费四个字连续随机量的生成-基于分布函数 1. 概率积分变换方法#xff08;分布函数#xff09;2. Python编程实现指数分布的采样 1. 概率积分变换方法#xff08;分布函数#xff09;
Consider drawing a random quantity X X X from a continuous probability distribution with … 连续随机量的生成-基于分布函数 1. 概率积分变换方法分布函数2. Python编程实现指数分布的采样 1. 概率积分变换方法分布函数
Consider drawing a random quantity X X X from a continuous probability distribution with the distribution function F F F. We know F F F is a continues nondecreasing function if F F F has an inverse F − 1 F^{-1} F−1, then Z F − 1 ( U ) ZF^{-1}(U) ZF−1(U), where U U U is a random quantity drawn from U ( [ 0 , 1 ] ) U([0,1]) U([0,1]), is a random quantity as desired. Indeed, P ( X ⩽ z ) P ( F − 1 ( U ) ⩽ z ) P ( U ⩽ F ( z ) ) F ( z ) , ∀ z ∈ R P(X \leqslant z)P\left(F^{-1}(U) \leqslant z\right)P(U \leqslant F(z))F(z), \forall z \in \mathbb{R} P(X⩽z)P(F−1(U)⩽z)P(U⩽F(z))F(z),∀z∈R
Example: Exponential distribution Exp ( 1 ) \operatorname{Exp} (1) Exp(1). Exp (1) has a probability density function: f ( z ) { e − z , z ⩾ 0 , 0 , z 0. f(z) \begin{cases}e^{-z}, z \geqslant 0, \\ 0, z0 .\end{cases} f(z){e−z,0,z⩾0,z0. Its distribution function is F ( z ) { 1 − e − z , z ⩾ 0 , 0 , z 0. F(z) \begin{cases}1-e^{-z}, z \geqslant 0, \\ 0, z0 .\end{cases} F(z){1−e−z,0,z⩾0,z0. We only need to concentrate on F ( z ) F(z) F(z) on [ 0 , ∞ ) [0, \infty) [0,∞), and have F − 1 ( z ) − log ( 1 − z ) . F^{-1}(z)-\log (1-z). F−1(z)−log(1−z). So F − 1 ( U ) − log ( 1 − U ) F^{-1}(U)-\log (1-U) F−1(U)−log(1−U) has a probability distribution Exp ( 1 ) (1) (1). Because 1 − U ∼ U ( [ 0 , 1 ] ) 1-U \sim U([0,1]) 1−U∼U([0,1]), we have − log U ∼ Exp ( 1 ) -\log U \sim \operatorname{Exp}(1) −logU∼Exp(1).
For a distribution function which does not have an inverse, we define a generalized inverse as the following: F − ( z ) inf { x ∈ R : F ( x ) ⩾ z } . F^{-}(z)\inf \{x \in \mathbb{R}: F(x) \geqslant z\} . F−(z)inf{x∈R:F(x)⩾z}.
2. Python编程实现指数分布的采样 Assignment: Sample a random quantity Z ∼ Exp ( λ ) Z \sim \operatorname{Exp}(\lambda) Z∼Exp(λ) for some λ 0 \lambda0 λ0. import numpy as np
import matplotlib.pyplot as plt# Parameter for the exponential distribution
lambda_value 0.5# Generate random quantity using CDF method
u np.random.rand(1000) # Uniform random numbers between 0 and 1
Z -np.log(1 - u) / lambda_value# Plot histogram
plt.hist(Z, bins30, densityTrue, alpha0.6, colorb, labelSampled Data)
plt.xlabel(Value)
plt.ylabel(Density)
plt.title(Histogram of Exponential Distribution (Generated using CDF))
plt.legend()
plt.grid(True)
plt.show()
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