33. If Ri denotes the random amount that is earned in period i, then i=1 i1Ri...

33. If Ri denotes the random amount that is earned in period i, then ∞

i=1 βi−1Ri , where 0 <β< 1 is a specified constant, is called the total discounted reward with discount factor β. Let T be a geometric random variable with parameter 1 − β

that is independent of the Ri . Show that the expected total discounted reward is equal to the expected total (undiscounted) reward earned by time T . That is, show that E



∞

i=1

βi−1Ri



= E



T i=1 Ri