From the memoryless property of Exponential Distribution function: Suppose E1,λ and E1,μ are independent, then; P [min{ E1,λ , E1,μ } > t] = P [E1,λ > t] . P [E1,μ } > t] = eˉλt . eˉμt = eˉ(λ + μ)t which shows that minimum of E1,λ and E1,μ is exponentially distributed. So, E1, λ1+ λ2+ λ3+……. λn = min { E1,λ1, E1,λ2, E1,λ3, ……….., E1,λn } Here, if we put λ = 1, then; E1, 1+ 2+ 3+……. n = min { E1,1, E1,2, E1,3, ……….., E1,n }
