Under a suitable rare-event limit, what continuous distributions link to Poisson counts and exponential waiting times?

By grouping raw measurements into defined intervals (bins) and tallying the number of observations falling into each bin, a histogram is constructed. The height or area of each bar represents the frequency or count for that interval, visually revealing the regions where data points cluster most densely.

Conditions: The data consists of numerical measurements.; Measurements are sorted into predefined intervals (bins).; The count of observations in each bin is recorded.

Sums of independent exponential waits with the same rate give gamma waiting times.

Conditions: The exponential waits are independent.; All exponential waits share the same rate.

The geometric distribution describes the number of failures before the first success, provided the trials are independent and the success probability remains fixed.

Conditions: Trials are independent.; Success probability is fixed.; The convention counts the number of failures before the first success.

Counting successes in a fixed block of trials gives a binomial variable, while counting failures before a fixed number of successes gives a negative binomial variable.

Conditions: Trials are independent with a fixed success probability.; The first scenario involves a fixed total number of trials.; The second scenario involves waiting for a fixed number of successes.