Geometric Distribution vs Negative Binomial Distribution

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

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

／var／log marcus chiu