= Strict record
A <strict record> in a sequence $X_1,X_2,\ldots$ occurs at index $n\geq2$ when $X_n>\max\{X_1,\ldots,X_{n-1}\}$. 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 $\sum_{j=1}^n1/j$ of <strict records> up to $n$. Ties or dependence can change the calculation; identical marginal <probability distributions> alone do not justify the symmetry argument.
Back to article page
