Interior peaks in a random sequence of readings

Question

Eight distinct temperature readings are recorded in a row in a completely random order. We only count "interior peaks": positions 2 through 7 (not the two ends) whose reading is higher than both of its immediate neighbors.

On average, how many interior peaks are there?

Accepted Answer

Only the 6 interior positions (positions 2 through 7) can be counted here, since the ends are excluded by definition. Put one indicator on each interior position. Fix an interior position. It has 2 neighbors, and among the 3 distinct readings of the position and its neighbors, the position holds the largest in 	frac13 of orderings: P(	ext{a given interior position is a peak}) = \frac13. By linearity over the 6 interior positions, E[	ext{interior peaks}] = 6 	imes \frac13 = 2. Takeaway. Restricting to interior positions removes the endpoint special case entirely: each qualifying position i…

Interior peaks in a random sequence of readings

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