A value repeatedly replaced by a smaller random one

Question

A number starts at 100. Each day it is replaced by a random value chosen uniformly between 0 and its current value. This happens on three consecutive days. What is the expected value after three days?

Accepted Answer

Each day's new value is uniform on [0, 	ext{yesterday's value}], so the days are nested. Condition day by day with the tower rule, using the fact that a uniform draw on [0, m] averages \frac{m}{2}. Let V0 = 100 and let V1, V2, V3 be the values after days 1, 2, 3. Given yesterday's value, today's average is exactly half of it: E[V{t} \mid V{t-1}] = \frac{V{t-1}}{2}, \qquad 	ext{so} \qquad E[Vt] = \frac{E[V{t-1}]}{2}. Apply this three times, starting from E[V0] = 100: E[V1] = \frac{100}{2} = 50, \qquad E[V2] = \frac{50}{2} = 25, \qquad E[V3] = \frac{25}{2} = 12.5. So the expected value after …

A value repeatedly replaced by a smaller random one

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