Statistics
1635 Project 2
The Poisson Distribution
The
Poisson distribution arises in connection with Poisson processes. It applies to
various phenomena of discrete nature (that is, those that may happen 0, 1, 2,
3, ... times during a given period of time or in a given area) whenever the
probability of the phenomenon happening is constant in time or space. Examples
of events that can be modelled as Poisson distributions include:
- The number of cars that pass through
a certain point on a road during a given period of time.
- The number of spelling mistakes a
secretary makes while typing a single page.
- The number of phone calls at a call
center per minute.
- The number of times a web server is
accessed per minute.
- The number of roadkill found per unit
length of road.
- The number of mutations in a given
stretch of DNA after a certain amount of radiation.
- The number of unstable nuclei that
decayed within a given period of time in a piece of radioactive substance.
- The number of pine trees per unit
area of mixed forest.
- The number of stars in a given volume
of space.
- The number of soldiers killed by
horse-kicks each year in each corps in the Prussian cavalry. This example
was made famous by a book of Ladislaus Josephovich Bortkiewicz (1868–1931).
- The distribution of visual receptor
cells in the retina of the human eye.
How does this distribution arise? – The law of rare events
The binomial
distribution with parameters n and λ/n, i.e., the
probability distribution of the number of successes in n trials, with
probability λ/n of success on each trial, approaches the Poisson
distribution with expected value λ as n approaches infinity. This
is sometimes known as the law of rare events.
Read the section in the text regarding the Poisson distribution.
Historically, a McDonald’s manager knows that cars arrive at the drive through at the rate of 2 cars per minute between the hours of 12 noon and 1 pm. Determine the following probabilities:
- Exactly 6 cars arrive between 12 noon and 12:05 pm
- Less than 6 cars arrive between 12 noon and 12:05 pm
- At least 6 cars arrive between 12 noon and 12:05 pm
Remember both the binomial distribution and the Poisson distribution provide theoretical probabilities, which give rise to expected results of an experiment. Comparing expected results of an experiment to the empirical results is at the very core of statistical inference. If the two are significantly different, we might conclude that the theoretical probabilities do not accurately describe the outcomes of the experiment.
A biologist performs an experiment in which 2000 Asian beetles are allowed to roam in an enclosed area of 1000 square feet. The area is divided into 200 subsections of 5 square feet each. After 30 days, the biologist and her assistants enter the roaming area and count the number of beetles in each subsection.
|
Number of beetles |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
|
Number of subsections |
3 |
7 |
14 |
18 |
19 |
30 |
30 |
21 |
20 |
15 |
8 |
6 |
4 |
2 |
2 |
1 |
- Determine if the beetles follow a Poisson distribution. This would be an indication that the beetles are evenly distributed.
The section in the text also mentions that the Poisson distribution can be used under certain conditions to approximate the binomial distribution.
Six percent of the human population is blood type O-negative. O-negative is unique because it can be donated to people of any blood type. Suppose there is a car accident and a victim is in immediate need of a blood transfusion. There are 110 bystanders at the accident.
- Use the binomial distribution to determine the probability that at least one of the bystanders is O-negative.
- If the conditions are satisfied (show whether they are or not), use the Poisson distribution to determine the probability that at least one of the bystanders is O-negative.