Ratio of Certain Dependent Random Variables

Question

Disclaimer: This problem is for my own understanding and not for a class in any way.

Greetings!

I am trying to solve the following problem but I am unsure how to proceed beyond what I have below. The trouble I am having is that $Z$ and $W$ and not independent of one another and so I am not sure how to address this dependency.

I have solved for $Z$ using the standard approaches for hierarchical distributions and so I should be good up to that point.

Any ideas on how to proceed?

Let $w < v$. We wish to solve

\begin{align} W \overset{iid}{\sim} U(0,w)\\ Z \sim U(W,v) \end{align}

solving for $Z$, we obtain

\begin{equation} Z \sim - \ln z I_{(0,z)}(z) \end{equation}

and so

\begin{equation} \frac{Z}{W} \sim ??? \end{equation}

Answer 1

Comment: To me, this is still not clearly stated. (In particular, I don't follow your last two displayed formulas.) Here is a speculative interpretation of a special case. First, you select $W \sim \mathsf{Unif}(0, 5).$ Then you select $Z \sim \mathsf{Unif}(W, 10).$ So $Z$ has support $(0,10),$ but is hardly independent of $W$. Thus, the distribution of $Z$ is a 'mixture' of uniforms with random left boundaries determined by $W.$

[A simplification would be to let $Z$ be a 50:50 mixture of $\mathsf{Unif}(2,10)$ and $\mathsf{Unif}(4,10)$.]

A quick simulation (in R) of what I describe is shown below. If this is the right interpretation, seeing a histogram of a million simulated $Z$'s might help you solve the problem. (The 'ramp' at the left is not linear.) If not, maybe you can try once more to explain what you want.

set.seed(429);  m = 10^6;  a = 5;  b = 10;  z = numeric(m)
for(i in 1:m) {
  w = runif(1, 0, a);  z[i] = runif(1, w, b) }
summary(z);  sd(z)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
 0.006075  4.558774  6.387860  6.245682  8.191052 10.000000 
[1] 2.320161

hist(z, prob=T, col="skyblue2")
abline(v=c(a,b), col="red", lwd=3, lty="dashed")

A scatterplot of the first 20,000 $(W,Z)$ pairs (captured by a sligtly different program) is shown below. Notice the higher density toward the right.