 Strict record

A strict record in a sequence $X_1,X_2,\dots$ occurs at index $n\ge 2$ when $X_n>\max \{X_1,\dots ,X_{n-1}\}$. The first observation is conventionally a strict record. A tie with the previous maximum is excluded: the observations $3,3,4,2,5$ have strict records at indices $1,3,5$.

If the observations are independent and identically distributed with a continuous cumulative distribution function, their maximum among the first $n$ observations is unique almost surely. By exchangeability, each of the $n$ positions has the same probability of containing that maximum. Thus the probability of a strict record at $n$ is $1/n$, and linearity of expectation gives an expected number ${\sum }_{j=1}^{n}1/j$ of strict records up to $n$. Ties or dependence can change the calculation; identical marginal probability distributions alone do not justify the symmetry argument.