Discrete random variable

Poisson distribution

The Poisson distribution 𝑋 ∼Pois(πœ†) describes the probability of π‘₯ events occurring in a fixed time interval, where πœ† is the expected number of occurrences and the time of each event is independent from any prior ones. prob It has the probability distribution

β„™(𝑋=π‘₯)=π‘’βˆ’πœ†πœ†π‘₯π‘₯!

Properties

  1. Expectation, variance: 𝔼⁑[𝑋] =Var⁑[𝑋] =πœ†
  2. Moment-generating function: 𝔼⁑[e𝑑𝑋] =eπœ†(eπ‘‘βˆ’1)
  3. Probability-generating function: 𝑔𝑋(𝑑) =eπœ†(π‘‘βˆ’1)

Additionally

  1. If 𝑋 ∼Pois(πœ†) and π‘Œ ∼Pois(πœ‡) are independently distributed then 𝑋 +π‘Œ ∼Pois(πœ† +πœ‡)
  2. If 𝑋 ∼Pois(πœ†π‘) and π‘Œ ∼Pois(πœ†π‘ž) where π‘ž =1 βˆ’π‘ are independently distributed then 𝑁 =𝑋 +π‘Œ ∼Pois(πœ†) and 𝑋 βˆ£π‘ =𝑛 ∼Bin⁑(𝑛,𝑝).
  3. Conversely, if 𝑁 ∼Pois(πœ†) and 𝑋 βˆ£π‘ =𝑛 ∼Bin⁑(𝑛,𝑝) then 𝑋 ∼Pois(πœ†π‘) and π‘Œ =𝑁 βˆ’π‘‹ ∼Pois(πœ†π‘ž) are independently distributed.

Poisson paradigm

Let {𝐴𝑖}𝑛𝑖=1 be independent events with 𝑝𝑖 =β„™(𝐴𝑖) small. Then 𝑋 =βˆ‘π‘›π‘—=1𝐼𝐴𝑗is approximated by 𝑁 ∼Pois(πœ†) where πœ† βˆ’βˆ‘π‘›π‘–=1𝑝𝑖, with

|β„™(π‘‹βˆˆπ΅)βˆ’β„™(π‘βˆˆπ΅)|≀min(1,1πœ†)π‘›βˆ‘π‘—=1𝑝2𝑗

for any 𝐡 βŠ†β„•. See also Poisson process.

Relationship to other distributions

  • If 𝑋 ∼Pois(πœ†) and π‘Œ ∼Pois(πœ‡) are independently distributed, then the conditional distribution of 𝑋 given 𝑋 +π‘Œ =𝑛 is Bin⁑(𝑛,πœ†πœ†+πœ‡).
  • As 𝑛 β†’βˆž and 𝑝 β†’0 as 𝑛𝑝 =πœ† remains fixed Bin⁑(𝑛,𝑝) ⇝Pois(πœ†).
  • By the Central limits theorem Pois(𝑛) ⇝N⁑(𝑛,𝑛) as 𝑛 β†’βˆž

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