In Casella and Berger, Statistical Inference, a problem is presented (4.61) involving the ratio $Z=\frac{X_2 - 1}{X_1}$, where $X_1$ and $X_2$ are independent exponential(1) random variables. I have computed the pdf of $Z$ in a couple of different ways, obtaining
$f_Z(z)=\frac{1}{e(z + 1)^2}$,
which is also the pdf reported in the solution's manual.
However, as far I understand, the support of $Z$ should be all real numbers, since $X_2 - 1$ can take negative values, and $X_1$ can get arbitrarily close to 0. But, if this is the case, then the above pdf of $Z$ has a discontinuity at $z=-1$ and the integral over its support doesn't converge.
What am I missing? I'm quite confused about this.
Hint: In your computation of $f_Z$ you have assumed that $1+zx_1 \geq 0$. This need not be true when $z <0$. So integration w.r.t $x_1$ is from $ 0$ to $-\frac 1 z$ when $z <0$ (and you have integrated from $0$ to $\infty$). Please go through your calculation with this point in mind and see if you get a genuine density function $f_Z$.