Example 4.6

Suppose a container contains an infinite collection of coins (may be biased or unbiased). Now whether the coin is biased or unbiased we still would have the probability lying between 0 and 1. Imagine you start flipping coins one after the other sequentially and our objective is to maximize the long run proportion of flips that land on heads.

Let us denote as the number of tails in the first  flips of the coins, so if  denotes the proportions of heads in the long run, then we must have the following . Let us reformulate the experiment in a way such that we achieve the same thing but in a different way. Let us choose the first coin and keep flipping the coin and at the same time we also keep counting the number of heads we get. We stop the time we get the first tail. Hence the moment the tail is obtained for the first time we can discard the first coin and pick up the second coin and keep a note that  is the number of heads obtained using the first coin. On a similar line we continue flipping the second coin till we get the first tail for the first time, i.e., the moment the first tail is obtained from the second coin we discard that coin and note the number of heads obtained by this second coin as . We continue doing this experiment, such that intuitively we have .

Suppose  and  and remember we now concentrate on the simple concept of  a geometric distribution of the form , where the mean is .
It is true that

,

thus with probability 1.

Example 4.7
One can easy proof the renewal equation

Example 4.8
We can show that the renewal function ,  uniquely determines the interarrival distribution .