Solutions - 5200-03

(1) Define $Z=X/Y$ and find the CDF of $Z$:

$$\begin{array}{c}P[Z\le z]=P[X/Y\le z]=P[X\le zY]\\ \\ \gg \gg {\int }_0^{\infty }{\int }_0^{zy}{e}^{-x}{e}^{-y}dxdy\\ \\ \gg \gg -{\int }_0^{\infty }{e}^{-y}{\left[e^{-x}\right]|}_0^{zy}dy\\ \\ \gg \gg 1-{\int }_0^{\infty }{e}^{-y(z+1)}dy\gg \gg 1-\frac{1}{z+1},z>0\end{array}$$

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(2) Differentiate to find $f_Z(z)$:

$$f_Z(z)=\frac{1}{{(z+1)}^2},z>0$$