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[null,"This is a tricky distribution for first year students and few will understand it"]
[null,"The key idea to get across here is the relationship with the Poisson"]
[null,"Thus, in this example, the relationship between #potholes and distance between potholes and the difference in the distributions is central"]
[]
[null,"Exercises"]
[null,1,"Change the rate of occurrence of potholes per km. (lambda) - understand how this affects the average inter-pothole distances"]
[null,2,"Change the number of bins (histogram intervals) - note that the exponential in unbounded on the right"]
[]
[]
[]
[]
[]
[]
[]
[null,"The authorities want to survey the state of the roads in SA before the 2010 World Cup"]
[null,"A company does a survey and ascertains that in the approximately 1 000 kms they survey "]
[null,"there are 1000 pot holes"]
[null,"However, all the information about where the potholes are located gets lost!"]
[]
[null,"If we assume that the potholes are in fact randomly distributed across the 1000 kms, we can approximate the "]
[null,"probability of different numbers of potholes on a randomly selected piece of road"]
[null,"In particular, we can answer questions of the type:"]
[null,"If I select a random 1 km (or 5 km) piece of road what is the probability of"]
[null,"0, 1, 2 or 15 potholes occuring in that road"]
[]
[null,"The simulation works as follows:"]
[null,"We get the computer to randomly distribute the locations of the 1000 potholes over the 1000 kms of road."]
[null,"by randomly allocating distances betwwen 0 and 1 000kms"]
[null,"We then view (consider) a random piece of road, say a randomly selected 2km stretch"]
[null,"and count the number of potholes in that section"]
[null,"We then repeat the sampling process of repeatedly taking 2km lengths and counting the number of potholes in the length"]
[null,"We then have assessed the empirical distribution of the number of potholes in a randomly selected 2km length of road"]
[]
[null,"We see this distribution is asymmetrical with a long right-hand tail"]
[null,"the more we repeat the sampling process,  the closer the results will approximate a Poisson distribution"]

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