Let $X_1$ and $X_2$ denote the survival times of two computer components. Assume that $X_1, X_2 \sim \text{Exp}(1)$. Find the distribution of the total survival time of the two components ($T=X_1+X_2$) and the distribution of the ratio of the survival time of the first component and the total surviving time of the two components($R=X_1/(X_1+X_2)$) by deriving the joint pdf of $T$ and $R$ and integrating out $R$ and $T$ to the marginal pdfs of $T$ and $R$. Identify the two distributions.
My professor told me the answer is $R \sim \text{Unit}[0,1]$ but it doesn't help me very much. Just shows me that I can't solve it.
Any help would be greatly appreciated!