= Negative covariance of empty and positive detector-interval counts
{title2=$\operatorname{Cov}(N_E,N_P)=-Np_Ep_P$}
The \b[negative <covariance> of empty and positive detector-interval counts] arises because a fixed number of measurement intervals is divided into mutually exclusive categories. Let $E_i,P_i$ indicate that interval $i$ is respectively physically empty or strictly positive. The same interval cannot satisfy both, so $E_iP_i=0$ and
$$
\operatorname{Cov}(E_i,P_i)=-p_Ep_P.
$$
For $N$ independent repetitions with common probabilities, the counts $N_E=\sum_iE_i$ and $N_P=\sum_iP_i$ therefore have
$$
\operatorname{Cov}(N_E,N_P)=-Np_Ep_P.
$$
The <joint probability of empty and strictly positive detector-width intervals> gives the underlying multinomial model. The sign describes category-count dependence, not physical antagonism between two detectors.
For an original boundary example, take $N=100$ and $p_E=p_P=0.025$. Each count has <variance> $100(0.025)(0.975)=2.4375$, while their <covariance> is $-0.0625$. The total tail-category count has <variance> $2.4375+2.4375-0.125=4.75$. Treating the two counts as independent would give $4.875$ instead. The <covariance> is needed even though the individual measurement repetitions are independent.
Conditioning on the total is important. If the number of repetitions is random and independent of the category assignments, the <law of total covariance> gives
$$
\operatorname{Cov}(N_E,N_P)=\bigl(\operatorname{Var}(N)-\mathbb E[N]\bigr)p_Ep_P.
$$
An independent total with a <Poisson distribution> has equal mean and <variance>, making this <covariance> zero; a more variable total can make it positive. This is a different sampling protocol, not a contradiction of the fixed-total formula.
Shared calibration drift is another change of model. Repetitions may then have dependent category indicators, and cross-repetition <covariance> terms must be added. <Overdispersion of detector-interval counts under shared drift> can make a simple multinomial analysis inadequate.
NIST's measurement guidance retains common influences in uncertainty propagation and requires assumptions for confidence-level interpretation.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3; §§6.2–6.3, Expanded Uncertainty]{ref} State whether the repetition total is fixed, how the categories are defined and which calibration influences are shared before interpreting their empirical count <covariance>.
Back to article page
