说我有两个随机变量:
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，我可以通过以下方式获取单个Beta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)那两个Beta的乘积又如何呢?我可以分析性地执行此操作吗? (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，请查看本文和此答案.
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)