$\begin{array}{l}{\text { Consider a renewal process with mean interarrival time } \mu . \text { Suppose that each }} \\ {\text { event of this process is independently "counted" with probability } p . \text { Let } N_{C}(t)} \\ {\text { denote the number of counted events by time } t, t>0 \text { . }} \\ {\text { (a) Is } N_{C}(t), t \geqslant 0 \text { a renewal process? }} \\ {\text { (b) What is } \lim _{t \rightarrow \infty} N_{C}(t) / t ?}\end{array}$
My Solutions
a)
Yes this is a renewal process with mean interarrival time, $p\times\mu$
b)
Using the elementary renewal theorem we know that $$\lim_{t\rightarrow\infty}\frac{N(t)}{t} = \frac{1}{\mu}$$
So
$$\lim_{t\rightarrow\infty}\frac{N_C(t)}{t} = \frac{1}{p\times\mu}$$
I was hoping someone could say whether this is correct?