6.5 Comparison of Distributions of Counts

The following table summarizes the four distributions we have seen that are used to model counting random variables. Note that Poisson distributions require the weakest assumptions.

Distribution	Number of trials	Number of successes	Independent trials?	Probability of success
Binomial	Fixed and known (\(n\))	Random (\(X\))	Yes	Fixed and known (\(p\)), same for each trial
Negative Binomial	Random (\(X\))	Fixed and known (\(r\))	Yes	Fixed and known (\(p\)), same for each trial
Hypergeometric	Fixed and known (\(n\))	Random (\(X\))	No	Fixed and known (\(p = \frac{N_1}{N_1+N_0}\)), same for each trial
Poisson	“Large” (could be random, could be unknown)	Random (\(X\))	“Not too dependent”	“Comparably small for all trials” (could vary between trials, could be unknown)