Probability and Statistics

Poisson Distributions

The probability mass function of Poisson distribution at x with parameter lambda is given by

pmf = dpois(x,lambda)

Similarly the cumulative distribution function is obtained by ppois(x,lambda).

lambda = 0.5
n = 15
x = 0:n
prob = dpois(x,lambda)
par(mfrow=c(2,1))
plot(c(-0.5,n+0.5),c(0,max(prob)+0.1),type='n',main='p.m.f',xlab='x',ylab='pmf')
prob.mass(x,prob,col='green')
x = c(-0.5,x,n+0.5)
cdf = ppois(x,lambda)
plot(x,cdf,type='s',col='red',main='c.d.f')
lines(c(-0.5,n+0.5),c(1,1),lty=2,col='red')
cdf

Programming Note. The function par(mfrow=c(2,1)) set the parameter mfrow=c(2,1) which splits the multiple graphics to assign 2 figures horizontally and one vertically.

Sample R code. You can download poisdemo.R, and run it.

Poisson approximation to binomial distribution. When n is large and p is small enough for np to become a moderate number, the binomial frequency function with parameter (n,p) can be approximated by that of the Poisson distribution with parameter $\lambda = np$ .

n = 30
p = 0.1
x = 0:n
bpmf = dbinom(x,n,p)
par(mfrow=c(1,1))
plot(c(-0.5,max(x)+0.5),c(0,max(bpmf)*1.1),type='n',
     main='Binomial distribution', xlab='x',ylab='pmf')
prob.mass(x,bpmf,col='green')

Here we can use a Poisson distribution as an approximation for a binomial distribution.

lambda = n*p
ppmf = dpois(x,lambda)
prob.mass(x,ppmf,col='red',density=10)
legend(x=0.7*n, y=max(bpmf)*1.1, legend=c('Binomial','Poisson'),
       col=c('green','red'),pch=c(15,22))

Sample R code. You can download poisapp.R, and run it.