I'm a bit confused about the lambda value of a Poisson distribution. I know it means the average rate of success for a given interval. I'm confused about what this value exactly means through.

For example, If I have 2.4/100,000 people contracting a disease over a period of two years, what is the lambda value if I'm trying to figure out the probability of at least 5 cases of the disease out of 100,000 in one year? I'm not sure if it would be either 2.4/100,000 or 1.2/100,000 since the original average is over a period of two years. I'm also pondering if lambda could be 2.4 or 1.2 since the question already states a sample of only 100,000.



Imagine we have a population of fixed size $100000$, say a small city. The "two year $\lambda$" is $2.4$. Then the "one year $\lambda$" is $1.2$.

You can think of it as follows. If the mean number of occurrences of the disease in a $2$ year period is $2.4$, then the mean number of occurrences in a $1$ year period is $1.2$.

Remarks: $1.$ Suppose we now change the city size to say $300000$. Then the appropriate $\lambda$ for a $1$ year period becomes $3.6$.

$2.$ For finishing your problem about at least five, it is easier to first find the probability that there are $4$ or fewer occurrences. This is $e^{-\lambda}\left(1+\lambda+\frac{\lambda^2}{2!}+\frac{\lambda^3}{3!}+\frac{\lambda^4}{4!}\right)$.

  • $\begingroup$ Last term should be 4!. I tried to edit it, but Stackexchange demands a minimum edit of 6 characters... :-/ $\endgroup$ – Pete Mancini Jun 15 '16 at 21:29
  • $\begingroup$ @PeteMancini: Thank you, I had missed the shift key. $\endgroup$ – André Nicolas Jun 16 '16 at 2:09

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