Poisson distribution

The Poisson distribution is a real discrete Probability distribution is the limit of the Binomial distribution where n→∞n\to \infty but the number of successes remains small (that is, np→νnp\to \nu). It describes rare events: events which have a low Probability of occurring even over many trials. It has a single parameter ν∈[0,1]\nu \in[0,1]. For a Random variable KK, the Probability mass function is

Pk=P(k;ν)=νke−νk!P_{k}=P(k;\nu)=\frac{\nu^{k}e^{-\nu}}{k!}

The Poisson distribution is commonly used for event counting. Suppose there are on average ν\nu independent events per unit time, and thus νΔt\nu\Delta t average total events in the time interval Δt\Delta t. The random variable KK is the number of events in Δt\Delta t, following the Poisson distribution. To prove this, divide the interval in nn sufficiently small, evenly-spaced steps δt=Δt/n\delta t=\Delta t/n such that the probability of two or more events occurring is approximately zero. Each one of the nn intervals has either zero or one event happen in it and the probability of having one event is given by the binomial distribution p=νδtp=\nu \delta t, with Expected value E[K]=νΔtE[K]=\nu \Delta t. If we add infinitely many intervals (n→∞n\to \infty) and send the probability of an event in each to zero (p→0p\to0) to keep their product constant, we get the Poisson distribution.

Moments

The raw moment-generating function is

MK∗(k)=∑k=1netkνke−νk!=e−ν∑k=0n1k!(νet)k=e−νeνet=e−ν(1−et)M_{K}^{*}(k)=\sum_{k=1}^{n} e^{tk} \frac{\nu^{k}e^{-\nu}}{k!}=e^{-\nu}\sum_{k=0}^{n} \frac{1}{k!}(\nu e^{t})^{k}=e^{-\nu}e^{\nu e^{t}}=e^{-\nu(1-e^{t})}

The central moment-generating function is

MK(t)=e−tνe−ν(1−et)=eν(et−1−t)M_{K}(t)=e^{-t\nu}e^{-\nu(1-e^{t})}=e^{\nu(e^{t}-1-t)}

Some moments are:

  • Raw 0. μ0∗=1\mu_{0}^{*}=1
    1. μ1∗=ν\mu^{*}_{1}=\nu (Expected value)
  • Central 0. μ0=1\mu_{0}=1
    1. μ1=0\mu_{1}=0
    2. μ2=ν\mu_{2}=\nu (Variance)
    3. μ3=ν\mu_{3}=\nu
    4. μ4=3ν2+ν\mu_{4}=3\nu ^{2}+\nu
  • Coefficients
    1. γ1=1/n\gamma_{1}=1/\sqrt{ n } (skewness, tends to become symmetrical when the number of sample events is very high)
    2. γ2=1/ν\gamma_{2}=1/\nu (kurtosis, tends to flatten the tails for a high number of events)

As limit of the binomial distribution

Let's consider the binomial distribution's MGF

MkB∗(t)=(etp+q)n=[(et−1)p+1]nM_{kB}^{*}(t)=(e^{t}p+q)^{n}=[(e^{t}-1)p+1]^{n}

and its expectation value E[k]=npE[k]=np. But np→νnp\to \nu in the Poisson distribution (so for n→∞n\to \infty), so we can write E[k]=νE[k]=\nu. Thus, p=ν/np=\nu/n. If we plug this in the MGF we get

MkB∗(t)=[(et−1)νn+1]nM_{kB}^{*}(t)=\left[ (e^{t}-1) \frac{\nu}{n}+1 \right]^{n}

which for n→∞n\to \infty tends to the Poisson MGF.

Relation to other distributions

It is the limit n→∞n\to \infty, p→0p\to0, np→νnp\to \nu of the binomial distribution. It also obeys the central limit theorem and tends to the Gaussian distribution for a large number of events.