A generalized Karhunen-Loève expansion

Let r (0 r 1) be the (unknown) reliability of a 'go-no go' machine. Let the discrete stochastic variable Sr be the number of successes out of n independent trials, given r (n is fixed). Define R (s) as follows: for every s R (s)= that r for which Pr(s, > s)= 0.1. R(s) (0 ' s ' n) gives the '90% one-sided confidence interval of r', so-called because if we perform n tests on a certain machine (whatever its reliability r is) and get s successes, then on the average the following assertion will be true in at least 90 out of 100 cases: