a) Cars arriving for gasoline at a Shell station follow a Poisson distribution w

Question

a) Cars arriving for gasoline at a Shell station follow a Poisson distribution with a mean of 10 per hour.

i) Determine the probability that over the next hour, only one car will arrive.

ii) Compute the probability that in the next 5 hours, more than 19 cars will arrive.

b) Complaints about an Internet brokerage firm occur at a rate of 7 per day. The number of complaints appears to be Poisson distributed.

i) Find the probability that the firm receives 5 or more complaints in a day.

ii) Find the probability that the firm receives 18 or more complaints in a 3-day period.

Explanation / Answer

PART 1
pmf of P.D is = f ( k ) = e- x / x!
where   
= parameter of the distribution.
x = is the number of independent trials
mean =
= 10
I.
P( X = 1 ) = e ^-10 * 10^1 / 1! = 0.0005
II.
mean rate for 5 hour period = 5 * 10
mean () = 50
P( X < = 19) = P(X=19) + P(X=18) + P(X=17) + P(X=16) + P(X=15) + P(X=14) + P(X=13) + P(X=12) + P(X=11) + P(X=10) + P(X=9) + ..P(X=0)
= e^-50 * 50 ^ 19 / 19! + e^-50 * 1 ^ 18 / 18! + e^-50 * ^ 17 / 17! + e^-50 * ^ 16 / 16! + e^-50 * ^ 15 / 15! + e^-50 * ^ 14 / 14! + e^-50 * ^ 13 / 13! + e^-50 * ^ 12 / 12! + e^-50 * ^ 11 / 11! + ......e ^-50 * 50^1 / 1!
e^-50 * ^ 10 / 10! + e^-50 * ^ 9 / 9! + = 0.00000048
P( X > 19) = 1 -P ( X <= 19) = 1 - 0 = 0.99999952

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