Properties of Poisson

1
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.
2
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
3
Example Binomial situation, n 100,
p0.075 Calculate the probability of fewer than
10 successes.
4
gt pbinom(9,100,0.075) 1 0.7832687 gt
This would have been very tricky with manual
calculation as the factorials are very large and
the probabilities very small
5
The Poisson approximation to the Binomial states
that ? will be equal to np, i.e. 100 x 0.075 so
?7.5
gt ppois(9,7.5) 1 0.7764076 gt
So it is correct to 2 decimal places. Manually,
this would have been much simpler to do than the
Binomial.
6
Poisson Approximation the Birthday Problem.
What is the probability that in a gathering of
k people, at least two share the same birthday?

7
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).
8
Then a simple conditional probability calculation
shows that pnk 1- p(all birthdays are
different)
We can write a simple R function - call it
probcoincide - to evaluate pnk for any n and k
9
gt probcoincide function(n,k) 1 -
prod((n-1)(n-k1))/n(k-1)
10
gt probcoincide function(n,k) 1 -
prod((n-1)(n-k1))/n(k-1)
gt probcoincide(365,22) 1 0.4756953 gt
11
gt probcoincide function(n,k) 1 -
prod((n-1)(n-k1))/n(k-1)
gt probcoincide(365,23) 1 0.5072972 gt
12
So that (on Earth) 23 is the minimum size of
gathering required for a better than evens chance
of two members sharing the same birthday. Proof
of this The mean number of birthday coincidences
in a sample of size k is
13
The number of birthday coincidences should have
an approximately Poisson distribution with the
above mean. Thus, to determine the size of
gathering required for an approximate probability
p of at least one coincidence, we should solve
14
In other words we are solving the simple
quadratic equation
In the case n365, p0.5, this gives k23.0
15
Simulation with Poisson
Just like in the case for Binomial, Poisson
results can be simulated in R. (rpois) Example Si
mulate 500 occurrences of arrivals at a bus-stop
in a 1 hour period if the distribution is Poisson
with mean 5.3 per hour.
16
gt ysimrpois(500,5.3) gt ysim 1 6 10 8 4 6
1 2 4 6 9 8 3 5 5 3 7 6 6 3 6 6
4 9 6 3 26 6 6 4 4 3 3 5 8 4 10 6
6 5 5 5 5 3 3 10 6 5 3 7 3 3 51
6 4 5 6 5 5 7 8 3 4 8 5 6 5 3 2 3
3 3 5 3 8 8 4 5 76 3 3 3 8 7 9
3 3 8 9 7 8 3 4 1 5 9 1 6 5 8 3 7
4 7 101 1 8 8 6 5 3 4 0 7 4 7 5
7 6 7 4 7 6 1 3 8 9 5 5 10 126 4 6
5 6 8 3 8 4 5 9 8 7 4 2 3 6 6 6
6 4 3 6 11 4 7 151 4 3 9 4 3 3 5 7
13 5 7 1 10 6 5 4 6 7 9 9 4 5 7 9
8 176 6 7 6 4 6 11 3 6 8 3 6 2 1 8
7 8 6 4 4 4 6 4 3 2 7 201 5 6 7
6 7 6 9 7 3 7 6 8 3 5 2 9 6 6 8 3
6 5 2 3 7 226 2 6 11 5 5 4 5 7 8
3 5 8 2 7 5 3 6 5 9 1 5 8 8 6
6 251 5 10 5 4 7 6 8 2 6 1 5 5 7 3
0 2 7 7 10 4 6 6 4 5 8 276 7 3 7
6 3 5 7 6 4 4 0 2 5 5 4 5 5 6 5 5
7 7 7 8 7 301 9 2 8 5 12 3 10 5 5
8 3 5 3 6 5 8 4 7 3 3 4 6 2 1
2 326 6 7 3 2 3 8 4 7 3 6 5 4 5 7
7 7 4 7 6 4 5 3 4 2 8 351 7 5 5
6 6 6 7 9 11 4 3 4 9 6 9 4 1 3 7 2
6 1 2 9 5 376 7 6 3 7 7 5 5 6 4
6 9 5 8 10 3 8 6 4 7 6 3 6 6 4
2 401 3 3 6 5 7 4 4 5 8 8 5 12 9 14
3 12 3 2 5 4 5 7 7 3 7 426 7 9 7
4 7 5 2 6 5 6 8 5 3 8 4 7 4 4 5 3
4 6 3 6 6 451 7 7 3 6 2 7 6 9 4
9 11 4 6 3 1 3 7 9 8 4 4 10 9 7
10 476 2 3 6 4 6 6 8 3 12 6 6 3 4
3 0 3 7 6 7 6 3 3 1 2 4
17
A table of the results is constructed
gt table(ysim) ysim 0 1 2 3 4 5 6 7
8 9 10 11 12 13 14 4 14 25 76 63 72 87 68
43 26 11 5 4 1 1 gt
18
A barplot can be drawn of the table
barplot(table(ysim))
19
Poisson distributions have expected value and
variance both equal to ?. Check this out for our
simulations.
gt mean(ysim) 1 5.44 gt var(ysim) 1 5.565531 gt
20
Both are slightly out so see what happens if we
simulate 5000 observations rather than 500.
gt ysimrpois(5000,5.3) gt mean(ysim) 1 5.3502 gt
var(ysim) 1 5.141388 gt
21
And for 50 000
gt ysimrpois(50000,5.3) gt mean(ysim) 1
5.29968 gt var(ysim) 1 5.335299 gt
22
R Packages
23
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.
24
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 gt
library(MASS)
25
To make available the dataset chem from within
MASS, use additionally gt data(chem) Documentatio
n on any package is available via the R help
system. Missing or further packages may usually
be obtained from CRAN.
26
Some data sets are already in R when you open it.
gt data(iris) gt iris Sepal.Length Sepal.Width
Petal.Length Petal.Width Species 1
5.1 3.5 1.4 0.2
setosa 2 4.9 3.0
1.4 0.2 setosa 3 4.7
3.2 1.3 0.2 setosa 4
4.6 3.1 1.5
0.2 setosa 5 5.0 3.6
1.4 0.2 setosa 6
5.4 3.9 1.7 0.4
setosa 7 4.6 3.4
1.4 0.3 setosa 8 5.0
3.4 1.5 0.2 setosa 9
4.4 2.9 1.4
0.2 setosa 10 4.9 3.1
1.5 0.1 setosa
27
Notice, though, that if you havent used the data
command, R will not know that iris exists.
Type demo()' for some demos, help()' for
on-line help, or help.start()' for a HTML
browser interface to help. Type q()' to quit
R. Previously saved workspace restored gt
iris Error Object "iris" not found gt
28
Similarly if you use a file from the library and
do not use the library command first, R will not
know that a data set exists.
Type demo()' for some demos, help()' for
on-line help, or help.start()' for a HTML
browser interface to help. Type q()' to quit
R. Previously saved workspace restored gt
data(chem) Warning message Data set chem' not
found in data(chem) gt
29
You also need to becoime familiar with the
command attach.
30
(No Transcript)
31
But if you attach iris