Expected number of record highs

Question

You observe n = 100 i.i.d. draws X1, X2, \dots, X{100} from a continuous distribution, in sequence. Call day i a record if Xi Xj for all j < i (day 1 is always a record). What is the expected number of records?

Accepted Answer

Let Ii indicate that day i is a record. Among X1, \dots, Xi, all i! orderings are equally likely (continuity rules out ties), so the maximum of the first i values is equally likely to sit at any of the i positions: P(Ii = 1) = \frac{1}{i}. By linearity of expectation, E[\text{records}] = \sum{i=1}^{100} \frac{1}{i} = H{100} \approx \ln 100 + 0.5772 \approx 5.19. Only about five records in a hundred days. In general the answer is the harmonic number Hn \approx \ln n + \gamma: records grow logarithmically, which is why "new all-time high" events get rare fast, the 1000-day count is only H{1000…

Expected number of record highs

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