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...

...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...

...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...

...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 animals is known to be 25%.In an experiment with 6 animals injected with a new vaccine it was observed that none of the animals caught the infection. Calculate the probability of observed result.
Q5. If on average 8 ships out of 10 arrive safely at a port, find the mean and standard deviation of the number of ships arriving safely out of a total of 1600 ships.
Q6. Eight coins are thrown simultaneously. Find the chance if throwing
(i) At least 6 heads.
(ii) No heads and
(iii) All heads
(iv)
Q7. In a town 10 accidents took place in a span of 50 days. Assuming that the number of accidents per day follows the Poissondistribution, find the probability that there will be three or more accidents in a day.
Q8. The distribution of typing mistakes committed by a typist is given below. Assuming a Poisson mode, find out the expected frequencies-
No. of mistakes per page | 0 | 1 | 2 | 3 | 4 | 5 |
No. of pages |...

...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...

...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 the probability
that a customer will spend more than 15 minutes in the bank given that he is still in the bank
after 10 minutes?
Solution:
P (X > 15) = e−15λ = e−3/2 = 0.22
P (X > 15|X > 10) = P (X > 5) = e−1/2 = 0.604
2
– Failure rate (hazard rate) function r(t)
r(t) =
f (t)
1 − F (t)
– P (X ∈ (t, t + dt)|X > t) = r(t)dt.
– For exponential distribution: r(t) = λ, t > 0.
– Failure rate function uniquely determines F (t):
F (t) = 1 − e
3
t
− 0 r(t)dt
.
2. If Xi, i = 1, 2, ..., n, are iid exponential RVs with
mean 1/λ, the pdf of n Xi is:
i=1
(λt)n−1
,
fX1+X2+···+Xn (t) = λe−λt
(n − 1)!
gamma distribution with parameters n and λ.
3. If X1 and X2 are independent exponential RVs
with mean 1/λ1, 1/λ2,
λ1
.
P (X1 < X2) =
λ1 + λ2
4. If Xi, i = 1, 2, ..., n, are independent exponential
RVs with rate µi. Let Z = min(X1, ..., Xn) and
Y = max(X1, ..., Xn). Find distribution of Z and
Y.
– Z is an...

...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...