I want to know the distribution (and the moments) of a variable, $Z = X/Y$, where $X\sim \mathcal{N}(\mu_{x}, \sigma^{2}_{x})$, and , $Y\sim \text{Lognormal}(\mu_{y},\sigma_{y})$? Hence, what I want is the distribution of $p(z|\mu_{x}, \sigma_{x}, \mu_{y},\sigma_{y})$.

Below there are two numerical illustrations of the wanted distribution. In the graphs is plotted:

  • In blue : ($\mu_{x},\sigma_{x},\mu_{y},\sigma_{y})' = (0, 1, 0, 1)'$
  • In red : ($\mu_{x},\sigma_{x},\mu_{y},\sigma_{y})' = (0, 15, 0, 1)'$
obs <- 10000
#### mu_x = 0 
X_0 <- rnorm(obs, 0 , 1) 
Y <- exp(rnorm(obs, 0 , 1)) 
Z_0 <- X_0 / Y
# 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
# 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)))

enter image description here


Finding a closed-form distribution for the product or ratio looks rather challenging, as even simple cases (zero mean and unit standard deviation) do not appear to produce closed forms.

The other part of the question ... to find the moments of $Z = X/Y$ ... is readily solvable, assuming that $X$ and $Y$ are independent random variables. In particular, if $Y\sim \text{Lognormal}(\mu_{y},\sigma_{y})$, and $W = 1/Y$, then $W\sim \text{Lognormal}(-\mu_{y},\sigma_{y})$. Then, by independence:

$$ E[Z^r] = E[X^r] \, E[W^r]$$

which is the product of the $r^\text{th}$ moment of a Normal random variable and the $r^\text{th}$ moment of a Lognormal random variable, both of which are standard results readily available in any textbook or wiki etc


The density of $Z$ is $$f(z)=\frac{1}{2\pi\sigma_x\sigma_y}\int_0^\infty \exp\{ -[zy-\mu_x]^2/2\sigma_x^2-[\log(y)-\mu_y]^2/2\sigma_y^2\}~\text dy$$ since the Jacobian of turning $(x,y)$ into $(zy,y)$ is $y$.

  • $\begingroup$ Any proof or reference to support this result? Thank you, $\endgroup$
    – POC
    Jun 4 at 14:07
  • $\begingroup$ Check jacobian and marginalization! $\endgroup$
    – Xi'an
    Jun 4 at 14:13
  • $\begingroup$ I am not an expert on this topic, so any help would be appreciated. $\endgroup$
    – POC
    Jun 4 at 14:35
  • 2
    $\begingroup$ It is not a matter of expertise: this one-line formula is the result of a change of variable from $(X,Y)$ to $(X,Z)$ and of a marginalization against $X$, hence follows from using standard probability tools. $\endgroup$
    – Xi'an
    Jun 4 at 16:59

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