Assignment (6)

Due (Individual and Group Parts): Apr. 3


In the first few weeks of this semester you learned about Monte Carlo Simulation, which simulates the real world events using a limited number of random samples of the event. This week we look more closely into the Markov Chains that is also based on random events. In brief, this is how Markov Chains (sequences) are made. In a perfectly random situation, a collection of random variables {xi} (where the index i runs through 0, 1, 2,..., n) having the property that, given the present, the future is conditionally independent of the past. A sequence that satisfies the following condition is called Markov. That is if X1, X2, X3, are completely random and if for any n, we can write:  

i.e., if the conditional distribution F of Xn assuming Xn-1, Xn-2, , X1 equals the conditional distribution F of Xn assuming only Xn-1. If a Markov sequence of random variates Xn take the discrete values a1, a2, a3, ..., an, then

and the sequence xn is called a Markov chain.



Here is what you need to do. Individually, flip a coin 50 times. For every Head choose move to the Right and for every Tail choose move to the Left. Draw the path you have obtained and compute your individual average. Then meet as group to compute yet another average for your group using the results from each member. Write a summary on the discussions of the results. You will submit the group meeting form and one individual form for the individual time you spent on the assignment.