Poisson Distribution
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Poisson distribution A Poisson process has the following characteristics: 1. It models the number of events x, that occur in an interval t, of a continuous medium such as time, space, or vol. 2. It is characterized by one parameter, l, the average number of events per unit interval of space or time 3. There is a constant and continuous probability of an event occurring per unit interval 4. The number of events that occur in any one interval is independent of the number that occurs in any other interval. It does not matter how far apart the events are in space or time. For instance, an event may have only just been observed or there may have been a considerable interval between them. 2. Average number events that occur per unit of exposure = λ 3. Random variable = X = number of events that occur in t units of exposure. 4. PM Func: Pr(x| λ,t) = ( λ t ) x e - λ t x! X = 0,1,2,… λt > 0 Mean number events in t units of exposure = λt Variance = λt e.g., mean = variance • Outcome is events in amount of “exposure”. Events do not have equivalent of failure and success. Event might be earthquakes in 1 year. No meaningful definition of “non-event”.
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Poisson Distribution
Transcript of Poisson Distribution
Poisson distribution
A Poisson process has the following characteristics:
1. It models the number of events x, that occur in an interval t, of a continuous medium such as time, space, or vol.
2. It is characterized by one parameter, l, the average number of events per unit interval of space or time
3. There is a constant and continuous probability of an event occurring per unit interval
4. The number of events that occur in any one interval is independent of the number that occurs in any other interval. It does not matter how far apart the events are in space or time. For instance, an event may have only just been observed or there may have been a considerable interval between them.
2. Average number events that occur per unit of exposure =
3. Random variable = X = number of events that occur in t units of exposure.
4. PM Func: Pr(x| ,t) = (t)xe-t x!X = 0,1,2,
t > 0Mean number events in t units of exposure = tVariance = t e.g., mean = variance Outcome is events in amount of exposure. Events do not have equivalent of failure and success. Event might be earthquakes in 1 year. No meaningful definition of non-event.
Medium of exposure is continuous (e.g., time, volume, etc.), not discrete like trial in Binomial.
No upper bound to random variable as in binomial.
The binomial distribution (j successes from n trials) is given by
Example
In a particular market discrete pieces of information arrive randomly at a rate of 10 per minute. What is the probability of only 8 pieces of information arriving in the next minute?. We could attempt to model this using a Binomial distribution assuming that the probability that the information arrives in the next second (n=60) is 10/60=1/6.
Of course we could object and say that we could have 2 pieces of information in a 1sec interval, hence we could chose n=120 (0.5s interval), p=10/120=1/12
We can continue along this path, n tends to infinity and p tends to zero with np=10.
The Poisson distribution is given by
The Poisson distribution is the limit of the binomial distribution as n, np= (the derivation is given below), where is the mean rate at which incidents occur.
For the above example =10, r=8
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