# London Underground

1. Probability Models and Normal Approximations
2. Engineering

There are a number of different subway lines in London, some of which run in parallel under the same streets. The builders of the Underground arranged this by stacking the subway tunnels for each line under each other, sometimes two or three deep. At many Underground stations, to get to the deepest tunnels you take a long escalator ride. For instance, at the Pimlico station the escalator down to the deepest tunnel is like a moving stairway with 96 steps. During rush hour every single escalator step typically has two people on it side by side, so that the escalator had to be designed to carry 192 people without overloading.

The population of Underground riders at rush hour is almost exclusively made up of adult men and women, whose weight averages about 150 pounds with an SD of about 28 pounds.

If the engineers who planned the Pimlico station designed the escalator to carry 29,700 pounds worth of people without breaking, what proportion of the time when it is fully loaded with 192 people would it break down? (Express your answer in the form "about 1 in every k fully loaded trips.")
Given that the morning and evening peak traffic periods together last about an hour and a half, and the turnover on the escalator is such that it's like having a new trip with 192 new people about every minute during this period, do you regard this failure rate as acceptably low? (Hint: At this rate about how often would it break down?)
If the engineers had wanted the escalator to fail only about once in every 10,000 trips (which would still mean it would overload about once every 110 days), how much weight should they have designed it to carry? (Be explicit about your probability model (in other words, relate this setup explicitly to the population-and-sample framework), and comment briefly on all assumptions you make and whether you think they are reasonable.)

Explain briefly.

George Michailides
gmichail@stat.ucla.edu