Beta–Geometric Waiting Time
See how uncertainty in one shared success probability creates decreasing hazards and a heavy waiting-time tail.
Free theory articleSee how uncertainty in one shared success probability creates decreasing hazards and a heavy waiting-time tail.
Concept & interpretation
The Beta–Geometric mixture models the waiting time until the first success when a single unknown success probability P is drawn once from a Beta(a, b) prior and held fixed across all trials of that experiment.
Interactive Experiment
Follow these steps in the Beta–Geometric Waiting Time module:
- Set Shape a to 2, Shape b to 2, and Trial index to 2.
- Observe that probePMF equals
1/5(0.2), survival equals1/2(0.5), and hazard equals2/5(0.4). - Compare the decreasing hazard curve with the constant hazard of the plug-in geometric model.
- Set Shape a to 1 to observe the boundary regime where
P(T ≥ t)decays asO(1/t)and the expected waiting time isPositive infinity.