Uniform Distributions Ratio

Question

Let $\xi$ is $U(0, 1)$, $\nu$ is $U(0, 1)$. What type of distribution is $g = \frac{\xi}{\nu}$? I have build logarithm of $g$, here is the plot but it seems like it is not normal distribution: Distribution Hist

Answer 1

Assming independence...

Consider the following system

$$\begin{cases} g=\frac{\xi}{\nu} \\ u=\xi \end{cases}$$

The Jacobian is $|J|=\frac{\xi}{g^2}$

thus your density is

$$f_G(g)=\frac{1}{2}\cdot\mathbb{1}_{[0;1)}(g)+\frac{1}{2g^2}\cdot\mathbb{1}_{[1;+\infty)}(g)$$

Answer 2

Since $\log 1/\nu$ and $\log 1/\xi$ are exponentially distributed, your plot, $\log g$, follows the Difference Between Exponential Distributions.

Answer 3

How did you receive the last statement? –

after calculating the jacobian that is $|J|=\frac{u}{g^2}$ you conclude that the joint density is

$$f(u,g)=\frac{u}{g^2}$$

but in the system you see that

$$0<\nu=\frac{u}{g}<1$$

Now, when $0<g<1$ this means that $0<u<g$ thus in this interval to get $f(g)$ you have to integrate the joint density in

$$f(g)=\int_0^g \frac{u}{g^2}du=\frac{1}{2}$$

When $g>1$ there are no problem for $\frac{u}{g}$ to be less than one thus you integrate

$$f(g)=\int_0^1 \frac{u}{g^2}du=\frac{1}{2g^2}$$

... put all together and get the result

this is the drawing of your density