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distribution of the total number of tumors produced in the whole population over a one-year period by this kind of radiation is 0, 1, 2, ..................104000.
let, x denote the total number of hits,
y denotes the number of hits by a particle in a year or an individual,
one per week ⇒ 52 per year
therefore, rate = 52 per year
since it is very rare to hit all individuals, we can use Poisson distribution.
therefore by using the formula p (x = x) = (e^(-rate) × rateˣ) ÷ x! ; x = 0, 1, 2, ............( 2000 × 52)
p (x = 2) = (e⁻⁵² × 52ˣ) ÷ x! ; x = 0, 1, 2, ............104000
now each hit has a harm of probability 10⁻⁵.
so p(total no. of tumors produced over a one year period)
= {e⁻⁵² × 52ˣ} ÷ x! × 10⁻⁵ ; x = 0, 1, 2, ............104000
therefore total no. of tumors = 2000 × (e⁻⁵² × ) ÷ x! × xⁿ ; x = 0, 1, 2, ............104000
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(complete question)
Assume that each of the 2000 individuals living near a nuclear power plant is exposed to particles of a certain kind of radiation at an average rate of one per week. Suppose that each hit by a particle is harmless with a probability of 1 - 10⁻⁵, and produces a tumor with a probability of 10⁻⁵. Find the approximate distribution of the total number of tumors produced in the whole population over a one-year period by this kind of radiation.