6.436/15.085  Fundamentals of Probability

final exam solutions

have been posted

Announced on 24 December 2007  6:13  p.m. by John Tsitsiklis

Grades etc.

The final exams have been graded and can be picked from my assistant, Lynne Dell. Grades for the semester will be available through Websis (or through email, if you ask).

Solutions will be posted in a few days.

Have a restful and joyful break
John Tsitsiklis

Technical postscript:

Several people tried to prove that the displaced process (in Problem 4) is Poisson by arguing, more or less, that "the arrival times are independent, so after a random displacement, they remain independent". Unfortunately, this type of argument is hard to translate to an actual proof.

One difficulty is that if we define S_i as the ith arrival time (so that these are actually *ordered* arrival times), once we displace them, their order may change; thus, the ith arrival time in the new process is actually a function of more than one arrival in the original process.

Another difficulty is that "independent arrival times" is not the same as the objective, i.e., proving that the number of arrivals in disjoint intervals are independent.

Direct proofs of the independence for different intervals are also very difficult to construct, much more difficult than the short proof that argues in terms of splitting and merging.

Announced on 19 December 2007  10:51  p.m. by John Tsitsiklis