We denote bynbspntnbspthe number of failures of a machine

Question

We denote by N(t) the number of failures of a machine in the interval [0, t]. We suppose that N(0) = 0 and that the time τ₀ until the first failure has a uniform distribution on the interval (0,1]. Similarly, the time τ_k-1 between the {k -l)st and the kth failure has a U(0,1] distribution, for k: = 2 , 3 , . . . . Finally, we assume that τ_o , τ₁ , . . . are independent random variables.