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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!

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1 Answer 1

Hints:

  1. You are given the distribution of $(X_1,X_2)$, this allows you to compute $\mathbb E(\varphi(X_1,X_2))$ for every bounded measurable function $\varphi$.
  2. You can deduce from this the value of $\mathbb E(\varphi(R,T))$ for every bounded measurable function $\varphi$.
  3. This yields the distribution of $(R,T)$.

If a step is unclear, say so.

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