Question related to equation in paper involving bernoulli process So I'm reading a paper about optimization of design of structures.
The author defines H(p) as the function that returns the cost of failure given decision variable p. So he computes the annual risk of failure as
$$H(p)*Pf$$
where Pf is the annual probability of failure.
Now he explores the case where after each failure the structure is built again and assumes that the Pf of each structure is an i.i.d. variable.
He then writes that the risk over infinite years as:
$$\ (1-Pf)*\sum_{n=1}^\infty (n*Pf^n)$$
where n is number of times failure occurs and structure is rebuilt.
Do you agree witn the above?
Shouldn't it be the binomial distribution of N failures over infinite years?
 A: It appears that the author is considering a process where the failed building is being rebuilt until it no longer fails, and after this the rebuilding process stops.
Given a fixed probability of failure $p_f$, the distribution of the number of trials (years) until the first success is Geometric, with expected value $p_f/(1-p_f)$.  Another way of writing this is as the author does: $(1-p_f)\sum_{n=1}^{\infty}np_f^n$.  This gives us the expected number of failures before the rebuilding process stops.  
Obviously the total cost is directly proportional to this number, since we pay $H(p)$ for each and every failure.  This is the basis for the statement that the "risk over infinite years is...". The "infinite years" part comes in because  there is no upper bound on the number of consecutive failures before the first (and only) success.  The expected cost inflicted on us next period, assuming we are attempting to build the building, is $H(p)p_f$, hence the "annual risk of failure statement".
