Compare three penalty schemes for a distribution of outcomes:
(a) Find the critical value of the standard variable and the average fine in euros.
(b) Find the corresponding critical value and average fine in euros.
(c) Analyse the scheme governed by a geometric distribution with p=q=21, and find the expected fine using
n=1∑∞4n(n−1)(21)n.
The sum can be evaluated by differentiating ∑n=1∞pn=p/(1−p) with respect to p.