／var／logmarcus chiu

Negative Binomial Distribution vs Binomial Distribution

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Negative Binomial (Pascal) Distribution

Binomial Distribution

  • 𝐏(𝑋=π‘₯; 𝑝,π‘˜) = 𝐏{ the π‘₯π‘‘β„Ž Bernoulli trial results in the π‘˜π‘‘β„ŽΒ success }
  • 𝐏(𝑋=π‘₯; 𝑝,π‘˜) = 𝐏{ (π‘˜-1) successes in the first (π‘₯-1) Bernoulli trials AND the last Bernoulli trial is success }
  • 𝐏(𝑋=π‘₯; 𝑝,π‘˜) = 𝐏{ (π‘˜-1) successes in the first (π‘₯-1) trials } 𝐏{ the last trial is success }

  • 𝐏(𝑋=π‘₯; 𝑝,π‘˜) = 𝐏(𝐗=π‘˜-1; 𝑝,𝑛=π‘₯-1) 𝐏(𝑋=π‘₯) # where:

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  	- 
  		- <font style="color: rgb(0,128,0);"><font style="color: rgb(255,102,0);">𝐏(𝐗=π‘˜-1; 𝑝,𝑛=π‘₯-1)<font style="color: rgb(122,134,154);">Β is a Binomial Distribution</font></font></font>
  		- <font style="color: rgb(0,128,0);">𝐏(𝑋=π‘₯)<font style="color: rgb(122,134,154);">Β is a Bernoulli Distribution</font></font>
  • 𝐏(𝑋=π‘₯; 𝑝,π‘˜) =Β [(π‘₯-1) choose (π‘˜-1)]Β (1-𝑝)π‘₯-π‘˜π‘π‘˜-1𝑝
  • 𝐏(𝑋=π‘₯; 𝑝,π‘˜) =Β [(π‘₯-1)Β chooseΒ (π‘˜-1)]Β (1-𝑝)π‘₯-π‘˜π‘π‘˜
  • 𝐏(𝑋=π‘₯; 𝑝,π‘˜) =Β [(π‘₯-1)!/[(π‘˜-1)!(π‘₯-π‘˜)!]]Β (1-𝑝)π‘₯-π‘˜π‘π‘˜
  • 𝐏(𝐗=π‘₯; 𝑝,𝑛) = 𝐏{ exactly π‘₯ success in 𝑛 Bernoulli trialsΒ }
  • 𝐏(𝐗=π‘₯; 𝑝,𝑛) = [𝑛 chooseΒ π‘₯] * prob-of-π‘₯-success * prob-of-(𝑛-π‘₯)-failures
  • 𝐏(𝐗=π‘₯; 𝑝,𝑛) = [𝑛!/(π‘₯!(𝑛-π‘₯)!] (1-𝑝)𝑛-π‘₯𝑝π‘₯

Let 𝑋=𝑛

  • 𝐏(𝑋=𝑛; 𝑝,π‘˜) = [(𝑛-1)!/[(π‘˜-1)!(𝑛-π‘˜)!]] (1-𝑝)𝑛-π‘˜π‘π‘˜
  • 𝐏(𝑋=𝑛; 𝑝,π‘˜) = (π‘˜/𝑛) 𝐏(𝐗=π‘˜; 𝑝,𝑛) # substitute Binomial Distribution

Let 𝐗=π‘˜

  • 𝐏(𝐗=π‘˜; 𝑝,𝑛) = [𝑛!/(π‘˜!(𝑛-π‘˜)!]Β (1-𝑝)𝑛-π‘˜π‘π‘˜
  • 𝐏(𝐗=π‘˜; 𝑝,𝑛) = (𝑛/π‘˜) 𝐏(𝑋=𝑛; 𝑝,π‘˜) # substitute Negative Binomial Distribution

Example Bernoulli trials for NBD π‘₯=10 and π‘˜=4:

  • πŸ”² ⬛️ πŸ”² ⬛️ πŸ”² ⬛️ πŸ”² πŸ”² πŸ”² ⬛️
  • πŸ”² πŸ”² ⬛️ ⬛️ πŸ”² πŸ”² ⬛️ πŸ”² πŸ”² ⬛️
  • ⬛️ πŸ”² πŸ”² ⬛️ ⬛️ πŸ”² πŸ”² πŸ”² πŸ”² ⬛️
  • πŸ”² πŸ”² ⬛️ ⬛️ ⬛️ πŸ”² πŸ”² πŸ”² πŸ”² ⬛️

NOTE: that every tenth box is black (e.g. success)

Example Bernoulli trials for BD 𝑛=10 and π‘₯=4:

  • πŸ”² ⬛️ πŸ”² ⬛️ πŸ”² ⬛️ πŸ”² πŸ”² ⬛️ πŸ”²
  • πŸ”² πŸ”² ⬛️ ⬛️ πŸ”² πŸ”² ⬛️ πŸ”² πŸ”² ⬛️
  • ⬛️ πŸ”² πŸ”² ⬛️ ⬛️ πŸ”² πŸ”² πŸ”² ⬛️ πŸ”²
  • πŸ”² πŸ”² ⬛️ ⬛️ ⬛️ ⬛️ πŸ”² πŸ”² πŸ”² πŸ”²

NOTE: that the tenth box doesn’t need to be black (e.g. success)