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...

...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 X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.
E(X) = S x P(X = x)
So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.
2. Define the following;
a) Binomial Distribution - is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Therewith the probability of an event is defined by its binomial distribution. A success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out...

...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 independent observations by the number of observations in the group that represent one of two outcomes. For example, the proportion of individuals in a random sample who support one of two political candidates fits this description. In this case, the statistic is the count X of voters who support the candidate divided by the total number of individuals in the group n. This provides an estimate of the parameter p, the proportion of individuals who support the candidate in the entire population.
The binomial distribution describes the behavior of a count variable X if the following conditions apply: the number of observations n is fixed, each observation is independent and represents one of two outcomes ("success" or "failure") and if the probability of "success" p is the same for each outcome.
If these conditions are met, then X has a binomial distribution with parameters n and p, abbreviated B(n,p).
* Bernoulli distribution
In probability theory...

...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 probability distribution problem. Each breakdown is called an occurrence in Poisson probability distribution terminology. The Poisson probability distribution is applied to experiments with random and independent occurrences. The occurrences are random in the sense that they do not follow any pattern, and, hence, they are unpredictable. Independence of occurrences means that one occurrence (or nonoccurrence) of an event does not influence the successive occurrences or nonoccurrences of that event. The occurrences are always considered with respect to an interval. In the example of the washing machine, the interval is one month. The interval may be a time interval, a space interval, or a volume interval. The actual number of occurrences within an interval is random and independent. If the average number of occurrences for a given interval is known, then by using the Poisson probability...

...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 above. Make sure to deﬁne all
terms in your deﬁnition.
(b) Suppose that 7000 claims are needed for full credibility (with range
parameter k = 0.1 and and probability level P ). Determine P .
1
2
2. A portfolio contains two types of risks: risk A and risk B:
• For risk A, the number of claims per year is independent and
follows a Poissondistribution with mean 1.
• For risk B, the number of claims per year is independent and
follows a Poissondistribution with mean 3.
Suppose that the portfolio contains the same number of people for each
risk. Consider a random insured individual and let Xj denote his claims
in year j.
(a) Calculate the probability that he makes k claims in the second
year (for k = 0, 1, . . .) given that he makes no claims in the ﬁrst
year.
(b) Find E[X2 |X1 = 0].
3
4
3. Suppose that:
• Conditioned on Θ = θ, the number of claims in each year N1 , N2 , . . .
are independent and Poisson distributed with parameter 3θ and
the claims amount in the jth...

...by 19 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 between exponential distribution and Poissondistribution?
II. Computation of probability (7%)
1. Let the random variable X follow a Binomial distribution with parameters n and p. We write X ~ B(n,p).
* Write down all basic assumptions of Binomial distribution.
* Knowing the p.m.f. of X, show that the mean and variance of X are = np, and 2 = np(1 – p), respectively.
2. A batch contains 40 bacteria cells and 12 of them are not capable of cellular replication. Suppose you examine 3 bacteria cells selected at random without replacement. What is the probability that at least one of the selected cells cannot replicate?
3. Redo problem No. 2 if the 3 bacteria cells are selected at random with replacement.
4. The number of customers who enter a bank in an hour follows a Poissondistribution. If P(X = 0) = 0.05, determine the mean and variance...

...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: University College Hospital (UCH), Ibadan, Nigeria. Methodology: A retrospective study involving a review of sickness records of all workers in UCH between January and December 2003 was carried out. Data were extracted from the staff records of the Staff Medical Services Department. Independent samples t-tests and one-way analysis of variance tests were used to test for statistically significant differences in the mean number of spells and duration between various groups of workers. Poisson regression models were fitted to describe the pattern of the number of spells of sickness. Results: Out of 3309 workers, 240 had records of sickness absenteeism, giving a prevalence rate of 7.3%. The mean spells of sickness was 3 spells per absentee per year, while the mean duration of absence was 4 days per absentee per year. Females had a significantly higher number of spells than males (p = 0.009) and longer duration of absence than males (p = 0.015). No statistically significant...

...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 discretedistribution (the Conway–Maxwell–Poissondistribution) is revived
and its statistical and probabilistic properties are introduced and explored. This distribution is a
two-parameter extension of the Poissondistribution that generalizes some well-known discrete
distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions
derived from these discrete distributions (i.e. the binomial and negative binomial distributions).
We describe three methods for estimating the parameters of the Conway–Maxwell–Poissondistribution. The ﬁrst is a fast simple weighted least squares method, which leads to estimates
that are sufﬁciently accurate for practical purposes. The second method, using maximum likelihood, can be used to reﬁne the initial estimates. This method requires iterations and is more...

...require that we know whether we have a sample or a population. 2. The following numbers represent the weights in pounds of six 7year old children in Mrs. Jones' 2nd grade class. {25, 60, 51, 47, 49, 45} Find the mean; median; mode; range; quartiles; variance; standard deviation. Solution: mean = 46.166.... median = 48 mode does not exist range = 35 Q1 = 45 Q2 = median = 48 Q3 = 51 variance = 112.1396 standard deviation =10.59 3. If the variance is 846, what is the standard deviation? Solution: 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 10 15 20
Solution: Weak positive linear correlation 6. What values can r take in linear regression? Select 4 values in this interval and describe how they would be interpreted. Solution: the values are between –1 and +1 inclusive. -1 means strong negative correlation +1 means strong positive correlation 0 means no correlation .5 means moderate positive correlation etc. 7. Does correlation imply causation? Solution: No.
8. What do we call the r value. Solution: The correlation coefficient....