5 There are on average 2 new cases each war of a rare diseas

5. There are on average 2 new cases each war of a rare disease in a given town. (a) What is the probability that on a given year there are 4 or more cases? b) What is the probability that on a 10 years period at least one year has 4 or more new cases?

Solution

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a)
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-2 * 2 ^ 3 / 3! + e^-2 * ^ 2 / 2! + e^-2 * ^ 1 / 1! + e^-2 * ^ 0 / 0!
= 0.8571
P( X > = 4 ) = 1 - P (X < 4) = 0.1429