Question

The development of events for an event based simulation, often requires the prediction of the time interval till the next event. To do this, many event simulations use a negative exponential distribution which is an approximation of the Poisson distribution. List two assumptions about events drawn from a Poisson distribution that make this approximation useful. Also give one real life example events that conform to the assumptions of the Poisson Proces. (25 points)

Answer 1

The poision distribution is derived as
p(k) = r*k / (k!)(e*r)

where r is average rate of occurrence and k is the number of times events

will be observed in a set of n units, calculate the kth term of the series to

get the probability of k, or p(k), and then multiply by the number n of units

in observation to get the number of probable total occurrences over the time

t.

The situations in which Poisson distributions apply are diverse, and it is

not always easy to see here examples of three common types of Poisson

problems

1 :Isolated Events

the average number of traffic accidents on the Hollywood Freeway between 7

and 8 PM on thursdayh mornings is 1 per hour. calculate the chance that there

will be 2 accidents on the Freeway, on some specific thursday morning

The basic rate is r = 1 in hour unit and our window is 1 hour. We wish to

know the chance of observing 2 events in that window. The rate r = 1 is

included in the Poisson Table, don't calculate anything. Read down the r = 1

column come to the p(2) row, and there the probability of 2 accidents is

0.1839 or a little less than 1 chance in 5.which is exact.

2: Proportions :like in calculating the amount of bacteria in a sample of

river water.

3:Arrivals:the number of calls a police station gets in an average for

calculating the crime rates.

some other real life examples where Poisson distribution is used like
How many emails a person will receive in a day,the Poisson distribution is

the limit of the Binomial distribution when n gets very large and p is very

small.

Poisson distribution is also used for modelling queuing system.for e.g.

people arriving at railway ticket counter. We can say that exponential

distribution gets implicitly used in this modelling i.e. the time between

arrivals of people at a ticket counter are exponentially distributed. Further

if we consider or note down the exact arrival times of these peoples then we

will find that they have gamma distribution.
Poisson process is also widely used in finance and risk management

distributions like t distributions are used.