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Assume we are are serviced by core $I$, where $I=i$ and $i=[0, n]$, with probability $p_i$. Also assume that the time needed by each $i$ in order to complete a job is an exponential random variable with parameter $\alpha_i$.

If $T$ is the time that a job will take in order to be completed, then what would $\mathbb{E}[T]$ and $\operatorname{Var}[T]$ be?

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$\newcommand{\E}{\mathbb{E}}$If I'm understanding your question correctly, these are the step-by-step solutions: $$\E[T]=\sum_{i=0}^n p_i \E[T|I=i]=\sum_{i=0}^n p_i α_i $$ and $$\operatorname{Var}[T]=\E[T^2]-\E^2[T]=\sum_{i=0}^n p_i \E[T^2|I=i]-\Big[\sum_{i=0}^n p_i α_i \Big]^2$$ $$=2 \sum_{i=0}^n p_i α_i^2 - \Big[\sum_{i=0}^n p_i α_i\Big]^2$$ Basically, you just use the total law of expectation, $\E[X] = \E[\E(X|Y)]$, along with the fact that $\E[X] = \alpha$, and $\E[X^2]=2 \alpha^2$ when $X$ is exponential.

Although I guess you have to be careful. There are two ways of parameterizing the exponential distribution and you didn't state which you're interested in. In the other parameterization, $\beta = 1/\alpha$.

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