Okay, that makes a lot of sense. I think I just confused interarrival times with just the number of arrivals.
For part
(c), I'm not sure what's going on.
(c) (i): If the number of arrivals
Xt is even, then I understand that we have one renewal every two arrivals, so therefore
Nt=21Xt.
Now if
Xt is odd, then since every second arrival is a renewal and our counting process for
Xt starts from 1, then we have 2, 4, 6... counting towards the renewal and if we land on say, 7, then we have
Nt=26=27−1=2Xt−1.
Is this a sufficient explanation?
For
(ii), I got:
m(t)=E(Nt)=21E(Xt)P(Xt Even)+21E(Xt−1)P(Xt odd)so
m(t)=21E(Xt)P(Xt Even)+21E(Xt)P(Xt odd)−21P(Xt odd)=21E(Xt)[P(Xt even)+P(Xt Odd)]−21P(Xt odd)And since P(Even)+P(odd) = 1, I get something close but not quite what I need. What am I doing wrong?
Edit: This does however give me exactly what I need for the last part. Perhaps it's an error?