Counting exceedances

A finite-sample quantity¶

For finite real observations $x_1,\ldots,x_n$ with $n\geq 1$, define the fraction strictly above a finite threshold $t$ by

The strict inequality matters when an observation equals the threshold. For $(1,2,2,4)$ at $t=2$, only the last observation contributes, so the fraction is $1/4$. Counting observations greater than or equal to 2 would give $3/4$ and answer a different question.

What the count establishes¶

Each indicator is either zero or one, so $0\leq\widehat S_n(t)\leq1$. If $s<t$, every observation exceeding $t$ also exceeds $s$. Therefore $\widehat S_n(t)\leq\widehat S_n(s)$. These statements follow directly from counting; they are exact finite-sample arguments.

The helper xmath.exceedance_fraction implements this convention. From a checkout, run uv run python demos/incerto/exceedances.py to obtain fractions 1, $1/4$, and 0 at thresholds 0, 2, and 4. The observations are synthetic and fixed.

A zero count above 4 does not establish that a population cannot exceed 4. Inference about an unseen population requires sampling and modeling assumptions that this example deliberately leaves unspecified. The Normix sample instead starts with an explicitly defined random-variable model.

This is an original worked derivation for the foundation, with no imported wiki text or external data.