Hi,

Could you please assist me in solving the following question? I'm getting a different answer for (a). I'm getting a wrong answer for (b) even after assuming that (a) is proved. I'm attaching the relevant slides from the coursework for your reference.

Consider a renewal point process with inter-arrival time distribution given by: F (t ) = 1-e3 - 3te 31, 120. a) Show that the renewal function R(t) has a LST R*(s) given by: R (s) =1+ 3 1 6 25 4s+ 6 where R" (s) is the LST of R(t)=1+m(t), which is the renewal function with R(0)=1. b) Conclude that the renewal function m(t) is given by: m (t) = (1-e- ), 120. c) Compute the following limits directly from (b): lim R(t +r) - R(t) = -, for any T> 0, m R (t ) lim 1 m im R (t ) - 1 m- + v 2m-The Renewal Function Definition: The renewal function of the process, m(t), is defined as the mean function of the counting process: m (t) = E (N.), tz0. We conclude (when we do not count an event occurrence at t=0): m ( 1 ) = EnP ( N, = n) = En [ F. ( 1 ) - F.. (1 ) ] = EF. (1). n=1 n=1 n=1 Thus, m(t) = EF. (1 ), 1 20. n=1 We note that at times we include an event occurrence at time 0 into the count; then, we denote the renewal function as R(t). We have: R(t) = > F,(t) = m(t) +1,t20; where we denote F (t) = F(t)" = U(1), n=0 as the unit step function U(t) that is =1 for t 2 0 and is =0 for t