6.2 Exercise 16

A sequence of trials (not necessarily independent) is performed. Let E_i be the event of success on the ith component trial. We associate with each trial a “payoff function” $X_{i} = a I_{E_{i}} + b I_{E_{i}^{c}}$. Thus, an amount a is earned if there is a success on the trial and an amount b (usually negative) if there is a failure. Let S_n be the number of successes in the n trials and W be the net payoff. Show that $W = \left(a - b\right) S_{n} + b n$.