Would I use method of transformation for this problem? Can someone help steer me in the right direction on this? I have the answer but no idea how to get to it.
Let X be a continuous random variable with density function
f(x) = 2/(x^2) for x ≥ 2 and 0 otherwise
Determine the density function of Y = X/(X−1) for 1 < y ≤ 2.
The way I am attempting to solve this is that I know x/(x-1) is decreasing. I am setting it up by method of transformation as P(x/(x-1) ≤ u)
\begin{eqnarray*} F_Y(y) &=& P(Y\leq y)\\ &=& P\left(\frac{X}{X-1}\leq y\right)\\ &=& P\left(X\leq y(X-1)\right)\\ &=& P\left(X\leq yX-y\right)\\ &=& P\left(y\leq yX-X\right)\\ &=& P\left(X \geq \frac{y}{y-1}\right)\\ &=& \int_{\frac{y}{y-1}}^{\infty} 2x^{-2}dx\\ &=& -2x^{-1}\Big|_{\frac{y}{y-1}}^{\infty}\\ &=& 2\frac{y-1}{y}\\ &=& 2(1-y^{-1}) \end{eqnarray*} Check: $F_Y(1)=0$ and $F_Y(2)=1$. All good!
Now, $f_Y(y)=F_Y'(y) = 2y^{-2}$ on $(1,2]$ and zero otherwise.