Properties of Poisson The mean and variance are both equal to . The sum of independent Poisson variables is a further Poisson variable with mean equal to the sum of the individual means. The Poisson distribution provides an approximation for the Binomial distribution.
Properties of Poisson. The mean and variance are both equal to . The sum of independent Poisson variables is a further Poisson variable with mean equal to the sum of the individual means. The Poisson distribution provides an approximation for the Binomial distribution. Approximation: - PowerPoint PPT Presentation
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Properties of Poisson
The mean and variance are both equal to .
The sum of independent Poisson variables is a further Poisson variable with mean equal to the sum of the individual means.
The Poisson distribution provides an approximation for the Binomial distribution.
Approximation:
If n is large and p is small, then the Binomial distribution with parameters n and p is well approximated by the Poisson distribution with parameter np,
i.e. by the Poisson distribution with the same mean
Example
Binomial situation, n= 100, p=0.075
Calculate the probability of fewer than 10 successes.
> pbinom(9,100,0.075)[1] 0.7832687>
This would have been very tricky with manual calculation as the factorials are very large and the probabilities very small
The Poisson approximation to the Binomial states that will be equal to np, i.e. 100 x 0.075
so =7.5
> ppois(9,7.5)[1] 0.7764076>
So it is correct to 2 decimal places. Manually, this would have been much simpler to do than the Binomial.
What is the probability that in a gathering of k people, at least two share the same birthday?
Poisson Approximation: the Birthday Problem.
Suppose there are n days in the year(on Earth we have n = 365)
Assume that each person has a birthday which is equally likely to fall on any day of the year, independently of the birthdays of the remaining k - 1 persons (no sets of twins in the group).
R is built from packages of datasets and functions. The base and ctest packages are loaded by default and contain everything necessary for basic statistical analysis. Other packages may be loaded on demand, either via the Packages menu, or via the R function library.
Once a package is loaded, the functions within it are automatically available. To make available a dataset from within a package, use the function data.
Of particular interest to advanced statistical users is the package MASS, which contains the functions and datasets from the book Modern Applied Statistics with S by W N Venables and B D Ripley. This package can be loaded with> library(MASS)
To make available the dataset chem from within MASS, use additionally
> data(chem)
Documentation on any package is available via the R help system.
Missing or further packages may usually be obtained from CRAN.