Poisson distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Poisson(μ) is used to indicate that the random variable X has the Poisson distribution with positive parameter μ. A Poisson random variable X with scale parameter μ has probability mass function

f(x) = \frac{μ^{x}e^{−μ}}{x!} x = 0,1,2,... .

The Poisson distribution can be used to model the number of events in an interval associated with a process that evolves randomly over space or time. Applications include the number of potholes over a stretch of highway, the number of typographical errors in a book, the number of customer arrivals in an hour, and the number of earthquakes in a decade. The Poisson distribution can also be used to approximate the binomial distribution when n is large and p is small. The probability mass function with μ = 8.56 is illustrated below.

PSfrag replacements

x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{Γ(x + 1, μ)}{Γ(x + 1)} x = 0,1,2,... .

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{Γ(x + 1) − xΓ(x, μ)}{x!} x = 0,1,2,... .

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{μ^{x}e^{−μ}}{Γ(x + 1) − xΓ(x, μ)} x = 0,1,2,... .

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = −ln[ \frac{Γ(x + 1) − xΓ(x, μ)}{x!} ] x = 0,1,2,... .

The inverse distribution function of X is mathematically intractable but can run in APPL with statement at the bottom of the page.

The median, m, of X is approximately (see Wikipedia site)

m ≈ ⌊μ + 1/3 − 0.02/μ⌋ .

The moment generating function of X is

M(t) = E [e^{tX}] = e^{μ(e^{t}−1)} − ∞ < t < ∞

The characteristic function of X is

ϕ(t) = E [e^{itX}] = e^{μ(e^{it}−1)} − ∞ < t < ∞

The population mean, variance, skewness, and kurtosis of X are

E[X] = μ  V [X] = μ  E [( \frac{X − μ}{σ} ) ^{3}] = μ^{−1/2}  E [( \frac{X − μ}{σ} ) ^{4}] = 3+μ^{−1} .

APPL verification: The APPL statements

X := PoissonRV(mu); CDF(X); SF(X); HF(X); CHF(X); IDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the cumulative distribution function, survivor function, hazard function, cumulative hazard function, inverse distribution function, population mean, variance, skewness, kurtosis, and moment generating function.