I'm working on this problem and I don't know where I got wrong:
Two independent random variables, $X\sim Uniform(0,1)$, $Y\sim Exp(1)$, find the distribution of $Z=\dfrac{X}{Y}$
To find the PDF, let $x=yz$, then $$f_Z(z)=\int_0^{\infty}zf_X(yz)f_Y(y)dy=\int_0^{\frac{1}{z}}ze^{-y}dy=z(1-e^{-\frac{1}{z}})$$
However, when I tried to find the CDF, I have $$P(Z\le z)=P(X\le yz)=P(X\le yz, yz\le 1)+P(yz> 1)=P(X\le yz, y\le\frac{1}{z})+P(y>\frac{1}{z})$$ $$=\int_0^{\frac{1}{z}}f_Y(y)\int_0^{yz}f_X(x)dxdy+\int_{\frac{1}{z}}^{\infty}f_Y(y)dy=z\int_0^{\frac{1}{z}}ye^{-y}dy+\int_{\frac{1}{z}}^{\infty}e^{-y}dy=z(1-e^{-\frac{1}{z}})$$
So I have its PDF is the same as CDF which is obviously wrong, but I don't know where. I think it's the PDF(actually it's from the solutions manual, I usually do CDF first) but I'm not sure. I've checked my calculation many times and I think that's not the problem.
Thanks in advance.