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Closed-form expressions for distribution of sum of exponential random variables

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2 Author(s)
Amari, S.V. ; Dept. of Ind. Eng. & Manage., Indian Inst. of Technol., Kharagpur, India ; Misra, R.B.

In many systems which are composed of components with exponentially distributed lifetimes, the system failure time can be expressed as a sum of exponentially distributed random variables. A previous paper mentions that there seems to be no convenient closed-form expression for all cases of this problem. This is because in one case the expression involves high-order derivatives of products of multiple functions. The authors prove a simple intuitive multi-function generalization of the Leibnitz rule for high-order derivatives of products of two functions and use this to simplify this expression, thus giving a closed form solution for this problem. They similarly simplify the state-occupancy probabilities in general Markov models

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Reliability, IEEE Transactions on  (Volume:46 ,  Issue: 4 )