A train arrives at a station such that the time between arrivals has a rate of exponential distribution of 1/λ. In order to get home, you decide to wait for the train for t minutes. If the train arrives before t minutes, you take the train home which takes time X. If the train has not arrived after t minutes, you walk home which takes time Y. What is the expected total time from getting to the train stop until getting home?

1. A computer train arrives punctually at a station every thirty-five minutes. Each morning, a commuter named Andrew leaves his house and casually strolls to the train station. Assume that the time he arrives at the station is random and uniform. Let X be the amount of time, in minutes, that Andrew waits for the train.

(a)[2] What kind of distribution is associated with the waiting time X? Give a graph to describe the shape of the distribution of X.  

A security and metal detection machine at an airport has a service rate that follows an exponential distribution with μ = 10 minutes.

Passengers arrive at the machine with an exponential rate of λ = 8 minutes. The queuing rule is FCFS. Find the expected machine utilisation, passenger throughput time, and average waiting time.

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store.

1. 1. Determine the expected number of customer arrivals for a five-minute period. (hint: this is   )
   2. If X = the number of arrivals per minute:
      Find the probability of 10 customers or less arriving within a minute.
   3. Find the probability of more than 7 customers arriving within a minute. 
   4. Find the probability of 3 customers arriving within a minute.