For which distribution same pdf is generated for given random variable 
For which of the distribution same pdf is generated for random variable X and 1/X. 
  Is it F(2,2)

 A: You are looking for a solution to $f(x) = \dfrac{f(1/x)}{x^2}$ 
So take any non-negative function $g(x)$ on $[-1,1]$ where $k= \int\limits_{-1}^{1} g(x)\,  dx$ is positive and finite 
then a solution will be    


*

*$f(x) = \dfrac{g(x)}{2k}  \text { when } -1 \le x \le 1$

*$f(x) = \dfrac{g(1/x)}{2kx^2} \text { when } x \lt -1 \text{ or } x \gt 1$
and I think all solutions will essentially be of this form
One solution is $f(x)=\dfrac{1}{(1+x)^2}$ for $x \gt 0$ and this is indeed an $F(2,2)$ distribution 
but there are many others, including the Cauchy density $f(x)=\dfrac{1}{\pi(1+x^2)}$ on $x \in \mathbb R$
Another simple illustration, with $g(x)=1$ and so  $k=2$, has $f(x)=\frac14$ when $x \in [-1,1]$ and $f(x)=\frac1{4x^2}$ otherwise.
A: Let $U,V$ be iid random variables with $P(U=0)=0$. 
Then if $X$ is defined as $\frac{U}{V}$ it will have the same distribution as $\frac1{X}=\frac{V}{U}$.
If moreover $X$ has a PDF then $\frac1{X}$ will also have the same PDF.
Special case: $U$ has chi-squared distribution. Then $X$ has $F$-distribution.
