Let $X_1,...X_n$ be a random sample from population with pdf $F(x|\theta)=\alpha \theta^{-\alpha}x^{\alpha -1}$ where $ 0 < x < \theta$
Show that $\frac{X_k}{X_{k+1}}$ and $\frac{X_l}{X_{l+1}}$ are independent if $k \neq l$. Find the pdf of $\frac{X_k}{X_{k+1}}$
So I haven't tried it yet because I wanted to make sure my method was valid. I figure first find the pdf of $\frac{X_k}{X_{k+1}}$, let's call it $f_k(x_k,x_{k+1})$ (the syntax my be off but just for example) by using a change of variables with the given pdf.
So for l, it should be the same except for the random variable. I basically want to show that $f_k*f_l=f_{k,l}$ to show indpendence. The left side would be easy, but I don't know how to construct the joint pdf to the right.
Or is there an easier way to go about this?