## 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) |