In a sequence of independent Bernoulli trials, the number of trials needed to obtain 𝑘 successes has Negative Binomial (Pascal) Distribution

Negative Binomial Distribution has two parameters:

• 𝑘 = number of needed successes
• 𝑝 = probability of success

With 𝑘=1, it becomes a Geometric Distribution

A Negative Binomial Variable 𝑿 can be represented as a sum of 𝑘 independent Geometric Variables (𝑋₁+ … + 𝑋_𝑘):

• 𝑿 = 𝑋₁+ … + 𝑋_𝑘

Probability Mass Function

• 𝐏(𝑋=𝑥) = 𝐏{ the 𝑥^𝑡ℎ Bernoulli trial results in the 𝑘^𝑡ℎ success }
• 𝐏(𝑋=𝑥) = 𝐏{ (𝑘-1) successes in the first (𝑥-1) Bernoulli trials AND the last Bernoulli 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) = 𝐏(𝐗=𝑘-1; 𝑝,𝑛=𝑥-1) 𝐏(𝑋=𝑥) # where:
  • 𝐏(𝐗=𝑘-1; 𝑝,𝑛=𝑥-1) - Binomial Distribution
  • 𝐏(𝑋=𝑥) - Bernoulli Distribution
• 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

Expectation

𝐄[𝑿] = 𝑘/𝑝

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the number of trials until the 𝑘^𝑡ℎ success is the summation of the following:

• Geometric number of trials 𝑋₁until the first success
• Geometric number of trials 𝑋₂until the next success
• …
• Geometric number of trials 𝑋_𝑘 until the next success

𝐄[𝑿] = 𝐄[𝑋₁ + … + 𝑋_𝑘] = 𝑘/𝑝

Variance

𝑉𝑎𝑟(𝑿) = 𝑘(1 - 𝑝) / 𝑝²

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• 𝑉𝑎𝑟(𝑿) = 𝑉𝑎𝑟(𝑋₁ + … + 𝑋_𝑘) # 𝑿 is a summation of geometric random variables
• 𝑉𝑎𝑟(𝑿) = 𝑉𝑎𝑟(𝑋₁) + … + 𝑉𝑎𝑟(𝑋_𝑘) # see variance properties: 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) + 2𝐶𝑜𝑣(𝑋,𝑌), covariance = 0
• 𝑉𝑎𝑟(𝑿) = [(1-𝑝) / 𝑝²] + … + [(1-𝑝) / 𝑝²]
• 𝑉𝑎𝑟(𝑿) = 𝑘(1 - 𝑝) / 𝑝²

Subpages

• Negative Binomial Distribution vs Binomial Distribution
• Geometric Distribution vs Negative Binomial Distribution

TODO

Negative Binomial Distribution #1

• the number of trials need to obtain k successes
• is the sum of k i.i.d. geometric distributions
• P(X=x) = (x-1 choose k-1) p^k (1-p)
• E[X]   = k/p
• Var[X] = k(1-p) / p
• M_X(t) = [ pe^t / (1 - qe^t) ]

Negative Binomial Distribution #2

• the number of failures before k successes
• P(X=x) = (x+k-1 choose x) (1-p)^x p
• E[X]   = k(1-p)/p
• Var[X] = k(1-p)/p
• M_X(t) = [ p / (1 - qe^t) ]