Suppose that the inter-renewal times of a renewal process have the (discrete) Poisson distribution with parameter 2.

Note that this means it is possible for successive renewal times to be the same. Find a series expression for the distribution of the number of renewals by time \(t\), and for the renewal function \(m(t)\). Produce a connected line graph of the large portion of the probability mass function of \(N_{6.5}\). Evaluate \(m(1)\) explicitly.