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Geometric Distribution vs Negative Binomial Distribution

Geometric Distribution vs Negative Binomial Distribution

Nov 08, 2021

Geometric Distribution

Negative Binomial (Pascal) Distribution

𝐏(𝑋=𝑥; 𝑝) = 𝐏{the 1^𝑠𝑡 success occurs on the 𝑥^𝑡ℎ bernoulli trial}
𝐏(𝑋=𝑥; 𝑝) = (1−𝑝)^𝑥−1𝑝

𝐏(𝑋=𝑥) = 𝐏{ the 𝑥^𝑡ℎ trial results in the 𝑘^𝑡ℎ success }
𝐏(𝑋=𝑥) = 𝐏{ (𝑘-1) successes in the first (𝑥 - 1) trials AND the last trial is success }
𝐏(𝑋=𝑥) = 𝐏{ (𝑘-1) successes in the first (𝑥 - 1) trials} 𝐏{ the last trial is success }
𝐏(𝑋=𝑥) = [(𝑥-1) choose (𝑘-1)] (1-𝑝)^𝑥-𝑘𝑝^𝑘-1𝑝
𝐏(𝑋=𝑥) = [(𝑥-1) choose (𝑘-1)] (1-𝑝)^𝑥-𝑘𝑝^𝑘
𝐏(𝑋=𝑥) = [(𝑥-1)!/[(𝑘-1)!(𝑥-𝑘)!]] (1-𝑝)^𝑥-𝑘𝑝^𝑘

𝐏(𝑋=𝑥; 𝑝) = (1−𝑝)^𝑥−1𝑝

𝐏(𝑋=𝑥; 𝑝,𝑘=1) = [(𝑥-1)!/[(1-1)!(𝑥-1)!]] (1-𝑝)^𝑥-1𝑝¹
𝐏(𝑋=𝑥; 𝑝,𝑘=1) = [(𝑥-1)!/(𝑥-1)!] (1-𝑝)^𝑥-1𝑝
𝐏(𝑋=𝑥; 𝑝,𝑘=1) = (1-𝑝)^𝑥-1𝑝