27. Let $\xi$ and $\eta$ be independent random variables having exponential distributions with parameters $\lambda$ and $\mu$ respectively. Find the distribution functions of the variables $\frac{\xi}{\xi+\eta}$ and $\frac{\xi+\eta}{\xi}$.
Solution. We have

$$
\frac{\xi+\eta}{\xi}=\frac{\lambda}{\mu}\left(\frac{\zeta+\eta}{\zeta}\right)-\frac{\lambda}{\mu}+1
$$

where $\zeta=\lambda \xi / \mu \stackrel{d}{=} \eta$. According to problem II.8.23(a), the quantity $(\zeta+\eta)^{-1} \zeta$ is uniformly distributed on the interval $[0,1]$. Therefore,

$$
\mathrm{P}\left(\frac{\xi+\eta}{\xi} \leqslant x\right)=1-\left(\frac{\mu}{\lambda}(x-1)+1\right)^{-1}
$$

and

$$
\mathrm{P}\left(\frac{\xi}{\xi+\eta} \leqslant x\right)=\left(\frac{\mu(1-x)}{\lambda x}+1\right)^{-1}
$$

in the case where the right-hand sides of the given equalities lie in $[0,1]$. For other values of $x$, the corresponding distribution functions are 0 or 1.