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A computer system has two independent processors, and functions as long as at least one processor has not failed. The times to failure of each processor are independent, each exponentially distributed with parameter $1$. Let $T_{1}$ be the time when the first processor fails, and let $T_{2}$ be the remaining time until the second processor fails, so the total life of the system is $T_{1} + T_{2}$.

  1. Find $\mathbb{E}\,T_{1}$

  2. Find $\mathbb{E}\,(T_{1}+T_{2})$

  3. What is $\mathrm{Cov}(T_{1},T_{2})$

  4. Find $\mathrm{Var}(T_{1}+T_{2})$

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Since the rvs are independent, their covariance is always 0, which answers the 3rd question. Since both variables are exponential, $ET_1= \mu=1$ ($\mu$ is the parameter of the rv). $\mathbf{E}(T_1 + T_2)=1+1=2, \ Var[T_1 + T_2]= 1+1=2$

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