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The Poisson distribution

Posted by Diego em Setembro 24, 2015


The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.


Example: let’s say that you are interested in knowing the probability that X cars  are going to pass in front your house in a given period of time.
To do that, you need to assume two things:


1)      Any period of time is no different than any other period of time;

2)      The number of cars that pass in one period doesn’t not affect the number of cars that will pass on the next period;



Give those assumptions, you need to find out how many cars on average pass in front of your house (that is your lambda  – λ).

If you want know the probability that K cars will pass, all you have to do is substitute λ and k on the formula bellow:





For example, lest say that the average is 9 cars and you want to know the probability that exactly 2 cars will pass:

λ =9



81 /2  * 0.000123   =  0.004998097


Or 0.5%



That same calculation can be can be achieved in R using the density dpois function:



[1] 0.004998097




To compute cumulative probabilities (for example, probability of 0 or 1 or 2 or 3 cars ), we can use the ppois function (which is the same as summing all individual probabilities with dpois):

[1] 0.02122649



Probability of having seventeen or more cars passing by your house:

ppois(16, lambda=9, lower=FALSE) #upper tail 
[1] 0.01110591


1-ppois(16, lambda=9) #lower tail
[1] 0.01110591




To finalise, we can simulate how many cars will pass on the next 10 hours:

[1]  9  9 11 15  7  8  8 13 11  9


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