Organisms are present in ballast water discharged from a shi

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms/m^3 [the article \"Counting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards\" (Ecological Applications. 2013: 339-351)]. What is the probability that one cubic meter of discharge contains at least 8 organisms? What is the probability that the number of organisms in 1.5 m^3 of discharge exceeds its mean value by more than one standard deviation?

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

Mean = Organisms are with a concentration of 10 Organisms/meter^3

a)
CONTAIN ATLEAST 8 ORGANISMS
P( X < 8) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-10 * 0 ^ 7 / 7! + e^-10 * ^ 6 / 6! + e^-10 * ^ 5 / 5! + e^-10 * ^ 4 / 4! + e^-10 * ^ 3 / 3! + e^-10 * ^ 2 / 2! + e^-10 * ^ 1 / 1! + e^-10 * ^ 0 / 0!
= 0.2202
P( X > = 8 ) = 1 - P (X < 8) = 0.7798