两个 beta 分布的乘积

本文介绍了两个 beta 分布的乘积的处理方法，对大家解决问题具有一定的参考价值，需要的朋友们下面随着小编来一起学习吧！ 问题描述

假设我有两个随机变量:

Say I have two random variables:

X ~ Beta(α1,β1)

X ~ Beta(α1,β1)

Y ~ Beta(α2,β2)

Y ~ Beta(α2,β2)

我想计算 Z = XY(随机变量的乘积)的分布

I would like to compute distribution of Z = XY (the product of the random variables)

使用 scipy，我可以获得单个 Beta 版的 pdf:

With scipy, I can get the pdf of a single Beta with:

from scipy.stats import beta rv = beta(a, b) x = np.linspace(start=0, stop=1, num=200) my_pdf = rv.pdf(x)

但是两个 Beta 的乘积呢?我可以分析吗?(Python/Julia/R 解决方案很好).

But what about the product of two Betas? Can I do this analytically? (Python/Julia/R solutions are fine).

推荐答案

对于分析解决方案，请查看 这篇论文和这个答案.

For an analytical solution, have a look at this paper and this answer.

R

set.seed(1) # for reproducability n <- 100000 # number of random variables # first beta distribution a1 <- 0.5 b1 <- 0.9 X <- rbeta(n, a1, b1) # second beta distribution a2 <- 0.9 b2 <- 0.5 Y <- rbeta(n, a2, b2) # calculate product Z <- X * Y # Have a look at the distributions plot(density(Z), col = "red", main = "Distributions") lines(density(X), lty = 2) lines(density(Y), lty = 2)