Fishing in Colorado for Poisson Parameters

Here’s a math problem.  Colorado State University’s Tropical Meteorology department does extensive research. It tells you that the probability of one or more named storms striking a particular Texas county during the 2013 hurricane season is x, where x is some value between 0 and 1.  They also tell you deep in their document that they are assuming that the distribution of named storm strikes tends to follow a “Poisson distribution.”  That’s a common assumption. What’s the probability of n strikes hitting that same Texas county during the 2013 hurricane season (where n is an arbitrary non-negative integer)?  This might be an interesting problem if you were trying to assess an insurance regulatory scheme, It could help you see whether that scheme provided adequate protection against multiple strike seasons.

Here’s some Mathematica code that solves this problem. First we fish — get it? — for the actual member of the Poisson family from which the relevant distribution is drawn.  We then refine our answer to take account of the fact that x must be less than 1. And then we wrap a PoissonDistribution function around our answer.

With[{soln =
Solve[SurvivalFunction[PoissonDistribution[k], 0] == x, k, Reals]},
PoissonDistribution[k /. First[Refine[soln, x < 1]]]]

From this, we learn that the associated Poisson Distribution has a parameter of -Log[1-x]. We can now compute the probability of n strikes as being …

The probability of n-strikes if the probability of at least 1 strike is x.

The probability of n-strikes if the probability of at least 1 strike is x.

 

 

 

 

And if you go over to my other blog, you can see this technique in action.

And here, by the way, is a non-math problem.  Why is a university about as far from being hit by a hurricane as an American university can be, a leading researcher in tropical cyclone risk?  Mathematica doesn’t provide an answer to that one.