My question is from George Casella statistical inference textbook Question 8.9 (b). I know how to do this question and I worked out. But I don't agree with the solution. In the question, it said "make the transformation $X_i = 1/Y_i$. Then I just replace $Y_i$ with $1/X_i$ in the original pdf $\lambda_i e^{-\lambda_i y_i}$. So my new pdf is just $\lambda_i e^{-\lambda_i/x_i}$.
However, the solution said the new pdf is $(\lambda_i/x_i^2) e^{-\lambda_i/x_i}$.
Does the solution have the typo? I used my pdf and I can solve out the problem.
The explanation forgets to include the Jacobian term. When in doubt, always go back to the explicit formula for performing change of variables. Assuming we have a transformation $X=g(Y)$ that is monotone on the support of $Y$ we can use the formula $$ f_X(x)=f_Y(g^{-1}(x))|J|, $$ where $|J|$ is the Jacobian of the transformation. In this particular case we have $X=1/Y$ (which is monotone on $\operatorname{supp}(Y)=(0,\infty)$) and so $g^{-1}(x)=1/x$ and $|J|=|\partial_x 1/x|=1/x^2$. Hence, $$ f_X(x)=f_Y(1/x)/x^2=(\lambda/x^2)\exp(-\lambda/x),\quad x>0, $$ which is the correct solution as given by the text.