Question

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms/m³ (the article "Counting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards"† considers using the Poisson process for this purpose).

(a)

What is the probability that one cubic meter of discharge contains at least 9 organisms? (Round your answer to three decimal places.)

(b) (THIS IS THE ONE I GOT WRONG)

What is the probability that the number of organisms in 1.5 m³ of discharge exceeds its mean value by more than two standard deviations? (Round your answer to three decimal places.)

(c)

For what amount of discharge would the probability of containing at least 1 organism be 0.995? (Round your answer to two decimal places.)

Answer 1

a)

this is Poisson distribution with parameter λ=10

P(X>=9)=1-P(X<=8)=

1-∑x=0x e-λ*λx/x!=

0.667

b)

this is Poisson distribution with parameter λ=10*1.5 =15

2 standard deviation from mean values are μ + 2*σ=22.75

P(X>=23)=1-P(X<=22)=

1-∑x=0x e-λ*λx/x!=

0.033

c)

amount of discharge =-ln(1-0.995)/10 =0.530 m³