Motor vehicles arrive at a toll gate according to a Poisson process with rate λ = 2 vehicles per minute. The vehicles arriving at the gate belong to classes 1, 2 or 3 with probabilities 1/2, 1/3 or 1/6, respectively. The drivers pay tolls of $1, $2 or $5 depending on the classes 1, 2, or 3 their vehicles belong to. (a) Find the probability that exactly $1 is collected in a period of 2 minutes. (b) Find the probability that the first toll collected is $1. (c) Find the probability that the waiting time between two vehicles that pay $5 is more than 10 minutes. (d) Find the mean and the variance of the amount in dollars collected in any given hour. (e) Suppose that 5 vehicles arrived at the toll gate in 10 minutes. Find the probability that only $5 dollars are collected and all are collected during the first 5 minutes.
Motor vehicles arrive at a toll gate according to a Poisson process with rate λ = 2 vehicles per minute. The vehicles arriving at the gate belong to classes 1, 2 or 3 with probabilities 1/2, 1/3 or 1/6, respectively. The drivers pay tolls of $1, $2 or $5 depending on the classes 1, 2, or 3 their vehicles belong to.
(a) Find the
(b) Find the probability that the first toll collected is $1.
(c) Find the probability that the waiting time between two vehicles that pay $5 is more than 10 minutes.
(d) Find the mean and the variance of the amount in dollars collected in any given hour.
(e) Suppose that 5 vehicles arrived at the toll gate in 10 minutes. Find the probability that only $5 dollars are collected and all are collected during the first 5 minutes.
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Motor vehicles arrive at a toll gate according to a Poisson process with rate λ = 2 vehicles per minute. The vehicles arriving at the gate belong to classes 1, 2 or 3 with probabilities 1/2, 1/3 or 1/6, respectively. The drivers pay tolls of $1, $2 or $5 depending on the classes 1, 2, or 3 their vehicles belong to.
(a) Find the probability that exactly $1 is collected in a period of 2 minutes.
(b) Find the probability that the first toll collected is $1.
(c) Find the probability that the waiting time between two vehicles that pay $5 is more than 10 minutes.
(d) Find the mean and the variance of the amount in dollars collected in any given hour.
(e) Suppose that 5 vehicles arrived at the toll gate in 10 minutes. Find the probability that only $5 dollars are collected and all are collected during the first 5 minutes.
please help with d and e
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