Organisms are present in ballast water discharged from a ship according to a Poi

Question

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms/m (the article "Counting at Low Concentrations: The Statistical challenges of Verifying Ballast Water Discharge Standardst considers using the Poisson process for this purpose). (a) What is the probability that one cubic meter of discharge contains at least 6 organisms? (Round your answer to three decimal places.) (b) What is the probability that the number of organisms in 1.5 m3 of discharge exceeds its mean value by more than one standard deviation? (Round your answer to three decimal places.) (c) For what amount of discharge would the probability ot containing at least 1 organism be 0.998? (Round your answer to two decimal places.)

Explanation / Answer

Here the process is poisson process so

= 10 organisms/m3

(a) Here if X is the number of organisms discharged

Pr(X 6 ) = POISSON (X   6 ; 10) = 1 - POISSON (X < 6 ; 10) = 1 - 0.067 = 0.933

(b) Expected number of organisms in 1.5 m3  of discharge = 10 * 1.5 = 15 organisms

Here standard deviation = sqrt(15) = 3.873

so as per the question,

Pr(15 - 3.873 < X < 15 + 3.873) = Pr (11.127 < X < 18.873) = Pr(X 19 ; 15) - Pr( X 11; 15) = 0.8752 - 0.1848 = 0.6904

(c) Here lets say the amount of discharge = x m3

Expected number of organisms = 10x

so,

Pr(X 1) = 1 - POISSON (X = 0; 10x) = 1 - e-10x = 0.998

e-10x = 0.002

10x = 6.2146

x = 0.6215 m3

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