Negative hypergeometric distribution

The negative hypergeometric distrubition $𝑋 \sim N H G e o m (𝑠, 𝑓, 𝑟)$ describes the probability of $𝑥$ failures occurring before the $𝑟$ -th success, drawn from a pool containing $𝑠$ successes and $𝑓$ failures, without replacement. #m/def/prob It has the probability mass function

ℙ (𝑋 = 𝑥) = (𝑥 - 𝑟 - 1 𝑟 - 1) (𝑠 + 𝑓 - 𝑟 - 𝑥 𝑓 - 𝑥) (𝑠 + 𝑓 𝑓)

Proof

#missing/proof See lecture 4 STAT2062

Properties

Let $𝑋 \sim N H G e o m (𝑠, 𝑓, 𝑟)$

$𝔼 [𝑋] = 𝑟 𝑓 𝑠 + 1$

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