What is the distribution of the reciprocal of a random variables with U[a,b]?

Question

Suppose that $Y_n$ are random variables that are uniformly distributed in the range $[a,b] (a>0)$.

What is the distribution of reciprocal $Z_n=\frac{1}{Y_n}$?

I found that it is Pareto distribution when $a=0$. But I found nothing when $a>0$.

Answer 1

We first find the cumulative density function $F_Z(z) = P(Z \leq z)$ and then differentiate it to obtain the probability density function $f_Z(z)$. Note that \begin{align*} F_Z(z) = P(Z \leq z) = P\left (\frac{1}{Y} \leq z \right) = P\left(\frac{1}{z} \leq Y \right) = 1 - P\left(Y < \frac{1}{z} \right) = 1 - F_Y\left( \frac{1}{z}\right), \end{align*} where $F_Y$ is the cumulative distribution function of $Y$. Then, we can write: $$\tag{1} F_Z(z) =1 - F_Y\left( \frac{1}{z}\right) = 1 - \int_{a}^{z^{-1}}\frac{1}{b-a}dx = 1 - \frac{z^{-1} - a}{b-a}.$$ If we set $a=0$ then we get Paretto cumulative density function. If we differentiate $F_Z(z)$ with $a=0$ we get Paretto probability density function $f(z) = \frac{b^{-1}}{z^2}$ with parameters $\alpha=-1$ and the scale parameter $b$.

Otherwise, just differentiate (1) and you obtain the probability density function $f_Z(z) = \frac{z^{-2}}{b-a}$.