Rate of events in a fixed time interval
On average five birds hit the Washington Monument and are killed each week. An official of the National Park has requested that congress allocate funds for equipment to scare birds away from monument. The congress committee replied that the funds cannot be allocated unless the probability of more than three birds being killed in any week exceeds 0.7. Will the funds be allocated?
Assume that the birds hit the monument at random, that the rate of collisions is constant over any given interval (here, 5 per week) and in particular that the expected time until a bird hits the monument (1/5th of a week) is independent of when the last bird hit the monument. These are the properties of the Poisson distribution.
Define `X` as the number of birds hitting the monument each week and assume `X` follows a Poisson distribution with rate `lambda=5` hits per week.
We write the probability density as
`Pr(X=x) = (lambda^x e^(-lambda))/(x!)` where `x` can equal 0,1,2,3, ....` `
The probability that `X > 3` equals 1 minus the probability that `X = 0, 1 ` or `2`, ie
`Pr(X>3) = 1 - Pr(X=0) - Pr(X=1) - Pr(X=2) `
`= `` 1 - (5^0 e^(-5))/(0!) - (5^1e^(-5))/(1!) - (5^2e^(-2))/(2!)`
`= 1 - e^(-5) - 5e^(-5) - 12.5e^(-5) = 1 - 0.125 = 0.875`
The probability is 0.875 which is larger than 0.7 so the funds will be allocated if the committee are convinced the model is accurate.