Poisson Distribution Question

Grasshoppers are distributed at random in a large field according to a Poisson distribution with parameter lambda = 2 per square meter. How large should a study section be (in square meters) so that the probability of finding at least one individual there is 0.99? Hint: here it is easier to reason with the probability of finding no grasshoppers...

How do I go about determining how large a study section should be?

we want that P[0] = 0.01

for that, if m is the mean *for the required area*,

e^-m *m^0/0! = e^-m = 0.01

taking natural logs, -m = ln 0.01, m = - ln 0.01 = 4.605

so area needed = 4.605/2 = 2.3025 m² <--------

Spence

Solve for λ ...

Note: Use the HINT by using P(X=0)

P(X = 0) = e^(-λ) (λ^0) / 0! = e^(-λ) = 0.1, now take the ln of both sides ...

ln[e^(-λ)] = ln 0.1

-λ = -2.30

λ = 2.30 square meter

hope that helped