Expected number of visible heads in a queue of 20

Question

Twenty people stand in a single-file line, their heights drawn i.i.d. from some continuous distribution. Looking from the front of the line, you can see a person's head exactly when they are taller than everyone standing in front of them. Let $X$ count the visible heads. Find $E[X]$ to one decimal place.

Short answer

About $3.6$ . Write $X$ as a sum of indicators: the $i$ -th person is visible iff they are the tallest of the first $i$ , which has probability $\tfrac1i$ . By linearity, $E[X]=H_{20}=\sum_{i=1}^{20}\tfrac1i\approx3.598$ .

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