Automobiles arrive at a vehicle equipment inspection station according to a Pois

Question

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate = 8 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations. (a) What is the probability that exactly eight arrive during the hour and all eight have no violations? (Round your answer to four decimal places.) (b) For any fixed y 2 8, what is the probability that y arrive during the hour, of which eight have no violations? (c) what is the probability that eight "no-violation" cars arrive during the next hour? [Hint: Sum the probabilities in part (b) from y-8 to.] (Round your answer to three decimal places.)

Explanation / Answer

Poisson Rate = 8 per hour

Pr(no violations) = 0.5

(a) As events "number of vehicles arrive in an hour" (x) & "violation with no violations" (y) are independent

Pr(8 comes in an hour and all have no violations) = POISSON(X = 8; 8) * BINOMIAL(y = 8 ;8; 0.5)

= 0.1396 * 0.0039 = 0.000545

(b) here EIght have no violations. so here we have to find that y arrive during the hour.

THis is a negativ binomial distribution.

Pr(y) = y-1C7 * 0.5y-8 0.58 = y-1C7 0.5y

(c) Here expected number of no violation to be occur in next hour = 8 * 0.5 = 4

Pr(Y = 8; 4) = POISSON ( X= 9 ; 4) = 0.0298

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