Poisson distribution | poisson distribution assumption
DiscreteprobabilitydistributionPoissonDistributionProbabilitymassfunctionThehorizontalaxisistheindexk,thenumberofoccurrences.λistheexpectedrateofoccurrences.Theverticalaxisistheprobabilityofkoccurrencesgivenλ.Thefunctionisdefinedonlyatintegervaluesofk;theconnectinglinesareonlyguidesfortheeye.CumulativedistributionfunctionThehorizontalaxisistheindexk,thenumberofoccurrences.TheCDFisdiscontinuousattheintegersofkandflateverywhereelsebecauseavariablethatisPoissondistributedtakesononlyintegervalue...
Discrete probability distribution
Poisson DistributionProbability mass function
The horizontal axis is the index k, the number of occurrences. λ is the expected rate of occurrences. The vertical axis is the probability of k occurrences given λ. The function is defined only at integer values of k; the connecting lines are only guides for the eye.
Cumulative distribution function
The horizontal axis is the index k, the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed takes on only integer values.
Notation Pois(λ){displaystyle operatorname {Pois} (lambda )}Parameters λ∈(0,∞){displaystyle lambda in (0,infty )} (rate)Support k∈N0{displaystyle kin mathbb {N} _{0}} (Natural numbers starting from 0)PMF λke−λk!{displaystyle {frac {lambda {k}e{-lambda }}{k!}}}CDFΓ(⌊k+1⌋,λ)⌊k⌋!,{displaystyle {frac {Gamma (lfloor k+1 floor ,lambda )}{lfloor k flo...