Discrete random variable

Negative hypergeometric distribution

The negative hypergeometric distrubition 𝑋 ∼NHGeom(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. prob It has the probability mass function

ℙ(𝑋=𝑥)=(𝑥−𝑟−1𝑟−1)(𝑠+𝑓−𝑟−𝑥𝑓−𝑥)(𝑠+𝑓𝑓)

Proof

proof See lecture 4 STAT2062

Properties

Let 𝑋 ∼NHGeom(𝑠,𝑓,𝑟)

1. 𝔼⁡[𝑋] =𝑟𝑓𝑠+1

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