Many studies are based on counts of the times a particular event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume, or any physical area in which there can be more than one occurrence of an event. Examples of variables that follow the Poisson distribution are the surface defects on a new refrigerator, the number of network failures in a day, the number of people arriving at a bank, and the number of fleas on the body of a dog. You can use the Poisson distribution to calculate probabilities in situations such as these if the following properties hold:

•You are interested in counting the number of times a particular event occurs in a given area of opportunity. The area of opportunity is defined by time, length, surface area, and so forth. •The probability that an event occurs in a given area of opportunity is the same for all the areas of opportunity. •The number of events that occur in one area of opportunity is independent of the number of events that occur in any other area of opportunity. •The probability that two or more events will occur in an area of opportunity approaches zero as the area of opportunity becomes smaller.

Consider the number of customers arriving during the lunch hour at a bank located in the central business district in a large city. You are interested in the number of customers who arrive each minute. Does this situation match the four properties of the Poisson distribution given earlier? First, the event of interest is a customer arriving, and the given area of opportunity is defined as a one-minute interval. Will zero customers arrive, one customer arrives, and two customers arrive, and so on? Second, it is reasonable to assume that the probability that a customer arrives during a particular one-minute interval is the same as the probability for all the other one minute intervals. Third, the arrival of one customer in any one-minute interval has...

...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...

...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...

...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:...

...Assignment
Q1Find the parameters of binomial distribution when mean=4 and variance=3.
Q2. The output of a production process is 10% defective. What is the probability of selecting exactly two defectives in a sample of 5?
Q3. It is observed that 80% of television viewers watch “Boogie-Woogie” Programme. What is the probability that at least 80% of the viewers in a random sample of five watch this Programme?
Q4. The normal rate of infection of a certain disease in...

...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...

...Exponential Distribution
• Deﬁnition: Exponential distribution with parameter
λ:
λe−λx x ≥ 0
f (x) =
0
x s).
=
=
=
=
=
P (X > s + t|X > t)
P (X > s + t, X > t)
P (X > t)
P (X > s + t)
P (X > t)
e−λ(s+t)
e−λt
e−λs
P (X > s)
– Example: Suppose that the amount of time one
spends in a bank is exponentially distributed with
mean 10 minutes, λ = 1/10. What is the probability that a customer will spend more than 15
minutes in the bank? What is...

...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...