During the time I lived in NYC, I used to take train to work every morning. It w

Question

During the time I lived in NYC, I used to take train to work every morning. It was taking 10 minutes for me to walk from my home to train station. Trains
used to arrived according to a Poisson process at a rate of 1 per 5 minutes. I had to take the train before 7:50 am to be able to get to work on time.
(a.) How many trains on average I used to miss during my walk from home to the station?
(b.) What was the latest time I should have left home each morning so that I could make it
to work on time at least 90% of the time?

Explanation / Answer

Ans:

a)As,there is 1 train per 5 min,so,on average you will miss 2 trains in 10 min.

2 trains

b)mean interarrival time between trains =5 min

interarrival time is exponentially distributed.

P(T<=t)=1-e-t/5

1-e-t/5>=0.9

t/5<=-ln(0.1)

t/5<=2.3

t<=11.5 minutes

It takes 10 minutes to walk and train should arrive within 11.5 min to reach him offie before time,so 10+22.5=32.5 min

(50-32.5=17.5

so,he should leave home by around 7:17 am

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