我想知道变量的分布（和矩） ，其中 和, ? 因此，我想要的是的分布。$Z = X/Y$$X\sim \mathcal{N}(\mu_{x}, \sigma^{2}_{x})$$Y\sim \text{Lognormal}(\mu_{y},\sigma_{y})$$p(z|\mu_{x}, \sigma_{x}, \mu_{y},\sigma_{y})$

下面是所需分布的两个数字说明。在图中绘制：

• 蓝色： ($\mu_{x},\sigma_{x},\mu_{y},\sigma_{y})' = (0, 1, 0, 1)'$
• 红色： ($\mu_{x},\sigma_{x},\mu_{y},\sigma_{y})' = (0, 15, 0, 1)'$

set.seed(123456L)
obs <- 10000
#### mu_x = 0 
X_0 <- rnorm(obs, 0 , 1) 
Y <- exp(rnorm(obs, 0 , 1)) 
Z_0 <- X_0 / Y
summary(Z_0) 
# Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
# -44.06862  -0.58312   0.01182   0.04968   0.62222  42.12307
hist(Z_0,breaks = 250,xlim = c(-10,50),  col='skyblue')

#### mu_x = 15
X_15 <- rnorm(obs, 15 , 1) 
Z_15 <- X_15 / Y
summary(Z_15) 
# Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
# 0.3174   7.5281  14.8337  24.5105  29.6927 657.0103 
hist(Z_10,breaks = 500,add = T,col=scales::alpha('red',.5))    
legend("topright", c("Z_0", "Z_15"), fill=c("skyblue", scales::alpha('red',.5)))