Cars and trucks arrive at a gas station randomly and independently of each other, at an average rate of 17.4 and 9.6 per hour, respectively.

Cars and trucks arrive at a gas station randomly and independently of each other, at an average rate of 17.4 and 9.6 per hour, respectively. Use the Poisson distribution to find the probability that

a. more than 5 cars arrive during the next 16 minutes,

b. we have to wait more than 21 minutes for the arrival of the third truck (from now),

c. the fifth vehicle will take between 17 and 23 minutes (from now) to arrive.

Solutions

Expert Solution

a) for 16 minutes

λ = 17.4 / hour * 16 min = 17.4 *16/60 = 4.64

P(X> 5) = 1- P(X<= 5) = 1 - 0.67885 = 0.32115

interarrival time is exponential distribution with parameter λ = 9.6/60

P(X <x) = 1 -e^(-λ x)

P(X> x) = e^(-λ x)

P(X > 21) = 1 - F(21) = e^(-21 * 9.6/60) = 0.03473525894

combined λ = 17.4/60 + 9.6/60 = 0.45

P(17 <X< 23)

=P(next vechile arrives<6mins)=

= 1-e^-.45*(6)=0.93279

orchestra answered 1 year ago

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