This problem set is due on Friday 3/13. Before working on it, you need to read Chapter 1 of Stochastic models, estimation, and control (volume 1) by Peter Maybeck, 1979. Feel very free to discuss it over email!
In addition to the following problem, please do problems 27, 29, and 31 from the probability textbook.
I am on a train from New Brunswick to New York, and wonder when I will arrive. To model the time a given train takes to get from one station to another, I assume a random variable X (the “sluggishness” of the given train), which is normally distributed with mean 1 and standard deviation 0.1. I then assume that, conditional on any given value of X, the three ratios
- time from New Brunswick to Rahway ÷ 20 minutes
- time from Rahway to Elizabeth ÷ 10 minutes
- time from Elizabeth to New York ÷ 30 minutes
are independently and normally distributed with mean X and standard deviation 0.05. Today the train to New York left New Brunswick at 6:49 pm, Rahway at 7:10 pm, and Elizabeth at 7:21 pm (all times are exact). When do I expect the train to arrive in New York?