Question

A critical part of a machine has an exponentially distributed lifetime with parameter α. Suppose
that n spare parts are initially at stock, and let N(t) be the number of spares left at time t.
(a) Find P(N(s + t) = j | N(s) = i).
(b) Find the transition probability matrix.
(c) Find Pj (t).

in Pj(t) j is in lower script

Answer 1

PM: Solution : Given that a part of a machine has an Exponentially distributed lifetime with paramete?'ar' let us assume that Q9umption : 'n' Spare parts are initially at stock, Murbeg of Spare parts left at time't' is nice) Here we have to find out ... a) tinding of_efo(s**) * - ;) ru (t) has reduced a part break down each time. and the time between break-dbone is distributed as ceponential with posameteq 'o'. let us Consider: Pos (1) . (+t). det = )

P.NO' = pſi-j break-downs in time 1] {:15j sisn'} (i-j) i probability density function of Exponential distribution . Coming to the initial state probability : =rli or more breate_dong as time t]... Since le jaisn}!... KOK! pro babiety density function of Exponential distribution.

دوP . N m. finding of transition probabilily matrix. [.ان. . r .. ا ز ابی [ ۲۰ : ۱۰ ۱۲۰,۰۰ .- از ۲۰۰۰۰) ۴- : ( ا ) - أ ) ع - ) ؟

pine: Transition probability Matrix : Koki (n-1)! (-2)! (not)

P. NO:5 a computation of P;(e let us Considen P; (t) = P n j (+) .:{{s jen} = (x+ya.cat (n-3), , probability deneity function of Exponentiae distribution . Coming to the initial Polle) = state probability , CE2 It is a probabitly density function of Exponential distribution.