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I have a question about probability, conditional expectations, copula functions and their use in a Monte carlo simulation. I'd appreciate any help or comments that you can offer.

I'll describe briefly what I am trying to simulate and I'll ask the specific question at the end.

I am trying to create a Monte Carlo simulation of the travel patterns of electric vehicles for a electricity grid impact analysis.

I have recorded the following variables for a fleet of cars over 6 months.

1. Journey Start Time

2. Journey End time

3. Journey Time

4. Journey Distance

5. The parking time between journeys

From the data above, I have extracted the following variables.

1. Departure time from Home (the first journey of the day)

2. Arrival Time Home (the last journey of the day)

3. The number of journeys made during the day (sum of the journeys)

4. The total distance travelled (sum of the individual journeys)

I found that these variables were correlated. I modelled the distributions of the variables and the dependence structure between them using a normal copula function. http://en.wikipedia.org/wiki/Copula_(probability_theory)

So the simulation begins by generating the 4 random variables above.

I also want to simulate the journeys during the day (this simulation above only generates the first and last journey time).

So in order to do this, I created two tables using all the data in the database as follows:

1. The Expected journey time given journey start time and journey distance. E(Journey Time | Start Time, Distance)

2. The Expected Parking time given journey stop time and the time already already parked during the day. E(Parking Time | Stop Time, Time already parked during the day)

The reason why I am using the "time already parked during the day" is because this prevents the parking time going over 24 hours.

This is a lot to take in, so I have made picture illustrating an example of the entire process.

https://dl.dropbox.com/u/54057365/All/pic%20sim.JPG

I've put my question in the green box. The question is theoretically speaking should the simulated arrival time home equal the arrival time home that is got by summing the expected journey times and parking time throughout the day with the simulated departure time in the morning?

I am finding that they are not equalling - I don't know if they should? My initial thoughts are that they probably won't because of the expected journey time and parking time tables.

I hope that I have explained this well enough.

I'd appreciate any help and comments.

Thanks

John

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# Question about a monte carlo simulation of car travel patterns

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