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Poisson Distribution Past Paper Question

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Hi,

Could anyone please explain how to solve this type of question:

In a ski resort, the probability of more than 3 accidents in any day at the weekend (Saturday and Sunday) is 0.24.

Assuming a Poisson model,

1) Calculate the mean number of accidents per day at the weekend (Saturday and Sunday)

  • I got to the point that P(x>3) =0.24 and P(x<3 or x=3) = 0.76.
  • How do I continue from here?

I also know that the answer is 2.49.

2) Calculate the probabilit that, in the four weekends in February, there will be more than 5 accidents during at least two of the weekends.

  • I am completely lost as to how to approach this question.

Any help (with clear explanations) is much appreciated.

Thanks!

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1. From the formula for Poisson distribution, you should solve via calculator that aEfFCW6.png. Note in calculator you can just directly plot the distribution. Eg. y = poissoncdf (x, 3) or whatever the syntax is and intersection with y = 0.76. 

2. I think they mean in each of at least 2 weekends, at least 5 accidents/weekend.
Because it's entire weekend, double expected value to 2.49 * 2 = 4.98.
Define a new poisson distribution Y ~ Po(4.98) and find 1 - P(Y=5) - P(Y=4) - P(Y = 3) - ... - P(Y=0) and say that's some probability p

Define a binomial distribution W ~ B(4, p) Question ask for P(W = 2) + P(W = 3) + P(W = 4). 

Hope that helps!

Edited by kw0573

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Ok thank you very much, I now managed to solve part 1.

For part 2 I am slightly confused about why you need both the poisson and the binomial distribution. Would you care to explain that further?

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2. The question asks for P(In at least 2 weekends, each weekend has at least 5 accidents). 

Poisson distribution is used when the number of favorable events can theoretically go up to infinity. Binomial distribution when the number of favorable events is bounded. 

Hence we first use poisson to calculate P(at least 5 accidents / weekend), as there can be as many accidents as possible in a weekend. Then we use binomial to calculate P(such weekends / month) since the number of weekends is limited to 4. 

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