The Poisson distribution is a discrete distribution. It is often used as a model for the number of events (such as the number of telephone calls at a business, number of customers in waiting lines, number of defects in a given surface area, airplane arrivals, or the number of accidents at an intersection) in a specific time period. It is also useful in ecological studies, e.g., to model the number of prairie dogs found in a square mile of prairie. The major difference between Poisson and Binomial distributions is that the Poisson does not have a fixed number of trials. Instead, it uses the fixed interval of time or space in which the number of successes is recorded.

Parameters: The mean is λ. The variance is λ.

[pic]

[pic] is the parameter which indicates the average number of events in the given time interval. Ex.1. On an average Friday, a waitress gets no tip from 5 customers. Find the probability that she will get no tip from 7 customers this Friday. The waitress averages 5 customers that leave no tip on Fridays: λ = 5. Random Variable : The number of customers that leave her no tip this Friday. We are interested in P(X = 7).

Ex. 2 During a typical football game, a coach can expect 3.2 injuries. Find the probability that the team will have at most 1 injury in this game. A coach can expect 3.2 injuries : λ = 3.2.
Random Variable : The number of injuries the team has in this game. We are interested in [pic].
Ex. 3. A small life insurance company has determined that on the average it receives 6 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.

P(x ≥ 7) = 1 - P(x ≤ 6) = 0.393697

Ex. 4. The number of traffic accidents that occurs on a particular stretch of road during a month follows a Poisson distribution with a mean of 9.4. Find the probability that less than two accidents will occur on...

...expected average outcome over many observations.The common symbol for the mean (also known as the expected value of X) is , formally defined by
Variance - The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.
Expectation - The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that...

...Binomial, Bernoulli and PoissonDistributions
The Binomial, Bernoulli and Poissondistributions are discrete probability distributions in which the values that might be observed are restricted to being within a pre-defined list of possible values. This list has either a finite number of members, or at most is countable.
* Binomial distribution
In many cases, it is appropriate to summarize a group of...

...The Poisson probability distribution, named after the French mathematician Siméon-Denis. Poisson is another important probability distribution of a discrete random variable that has a large number of applications. Suppose a washing machine in a Laundromat breaks down an average of three times a month. We may want to find the probability of exactly two breakdowns during the next month. This is an example of a Poisson...

...AMA470 Midterm exam
March 5, 2010
Please show full working out in order to obtain full marks.
1. Suppose that:
• The number of claims per exposure period follows a Poissondistribution with mean λ = 110.
• The size of each claim follows a lognormal distribution with parameters µ and σ 2 = 4.
• The number of claims and claim sizes are independent.
(a) Give two conditions for full credibility that can be completely
determined by the information...

...April 2013.
SPECIAL DISTRIBUTIONS
I. Concept of probability (3%)
1. Explain why the distribution B(n,p) can be approximated by Poissondistribution with parameter if n tends to infinity, p 0, and = np can be considered constant.
2. Show that – and + are the turning points in the graph of the p.d.f. of normal distribution with mean and standard deviation .
3. What is the relationship...

...ANALYSIS OF SICKNESS ABSENCE USING POISSON REGRESSION MODELS David A. Botwe, M.Sc. Biostatistics, Department of Medical Statistics, University of Ibadan Email: davebotwe@yahoo.com
ABSTRACT Background: There is the need to develop a statistical model to describe the pattern of sickness absenteeism and also to predict the trend over a period of time. Objective: To develop a statistical model that adequately describes the pattern of sickness absenteeism among workers. Setting:...

...
A useful distribution for ﬁtting discrete data: revival
of the Conway–Maxwell–Poissondistribution
Galit Shmueli,
University of Maryland, College Park, USA
Thomas P. Minka and Joseph B. Kadane,
Carnegie Mellon University, Pittsburgh, USA
Sharad Borle
Rice University, Houston, USA
and Peter Boatwright
Carnegie Mellon University, Pittsburgh, USA
[Received June 2003. Revised December 2003]
Summary. A useful discrete...

...standard deviation = square root of variance = sqrt(846) = 29.086 4. If we have the following data
34, 38, 22, 21, 29, 37, 40, 41, 22, 20, 49, 47, 20, 31, 34, 66 Draw a stem and leaf. Discuss the shape of the distribution. Solution: 2 3 4 5 6 | | | | | 219200 48714 0197 6
This distribution is right skewed (positively skewed) because the “tail” extends to the right. 5. What type of relationship is shown by this scatter plot?
45 40 35 30 25 20 15 10 5 0 0 5...

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