Comparing:

The binomial distribution tends toward the Poisson distribution as 𝑛→∞, 𝑝→0, and 𝑛𝑝→constant≈𝜆.

The Poisson distribution with 𝜆=𝑛𝑝 closely approximates the binomial distribution if 𝑛 is large and 𝑝 is small.

Derive Poisson Formula From Binomial PMF

𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛 choose 𝑥]𝑝^𝑥(1 - 𝑝)^𝑛-𝑥
𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑥!(𝑛-𝑥)!]𝑝^𝑥(1 - 𝑝)^𝑛-𝑥 # by binomial coefficient
𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑥!(𝑛-𝑥)!](𝜆/𝑛)^𝑥(1 - (𝜆/𝑛))^𝑛-𝑥 # 𝑝=𝜆/𝑛 when 𝑛→∞, 𝑝→0
𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑥!(𝑛-𝑥)!](𝜆/𝑛)^𝑥(1 - (𝜆/𝑛))^𝑛(1 - (𝜆/𝑛))^-𝑥 # by algebra
𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑥!(𝑛-𝑥)!](𝜆/𝑛)^𝑥𝑒^-𝜆(1 - (𝜆/𝑛))^-𝑥 # (1 - (𝜆/𝑛))^𝑛= 𝑒^-𝜆 as 𝑛→∞, see number e (Euler's number)
𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑥!(𝑛-𝑥)!](𝜆/𝑛)^𝑥𝑒^-𝜆(1 - (0))^-𝑥 # 𝜆/𝑛= 0 as 𝑛→∞
𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑥!(𝑛-𝑥)!](𝜆^𝑥/𝑛^𝑥)𝑒^-𝜆 # by algebra
𝐏(𝑋=𝑥) = 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑛^𝑥(𝑛-𝑥)!](𝜆^𝑥/𝑥!)𝑒^-𝜆 # by algebra
𝐏(𝑋=𝑥) = (𝜆^𝑥/𝑥!)𝑒^-𝜆 # 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑛^𝑥(𝑛-𝑥)!] = 1
1. Click here to expand...
  - 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑛^𝑥(𝑛-𝑥)!]
  - 𝑙𝑖𝑚_𝑛→∞ [𝑛!/(𝑛-𝑥)!]*[1/𝑛^𝑥]
  - 𝑙𝑖𝑚_𝑛→∞ [(𝑛-0)(𝑛-1)(𝑛-2)...(𝑛-𝑥+1)] / [𝑛^𝑥]
  - 1

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