Monthly Archives: October 2016

Probability Foundations – Problem (36/365)

I did say it would be one post a day and it already looks like i’ll only achieve it as an expectation of posts every day. So, let me catch up first by solving more problems. This time from chapter … Continue reading

Posted in Uncategorized | Tagged , | Leave a comment

Expectation Maximization – Link (35/365)

Dan Piponi has written up a simple to follow derivation of the Expectation-Maximization algorithm. It give a very practical derivation of the algorithm which also makes it easy to remember. What it clarifies for me is the step in the … Continue reading

Posted in Uncategorized | Tagged , | Leave a comment

Random Walk – Reading And Recall (34/365)

I said I wanted some awkward examples of random walks. After trying to shoehorn situations into a or movement, I think I have something. Consider the reading of a book. It’s an activity that proceeds in sequence as we read … Continue reading

Posted in Uncategorized | Tagged , , | Leave a comment

Random Walk (33/365)

One of the annoying things, for me, when studying probability are the examples furnished in textbooks because they almost always have to do with physics or gambling or finance. There is nothing wrong with physics examples except for the fact … Continue reading

Posted in Uncategorized | Tagged , | Leave a comment

Markov Chains – Problem (32/365)

Let be a stochastic matrix (i.e. and for all ). Show that is an eigenvalue of this matrix and all its eigenvalues satisfy . Since, is a stochastic matrix we can apply the Ergodic theorem which tells us that there … Continue reading

Posted in Uncategorized | Tagged , | Leave a comment

Markov Chains – Problem (31/365)

Going to work on a few more problems before I take a look at Random Walks, Martingales, and Markov chains together as they share some things in common as presented in the book. Let be a Markov chain with values … Continue reading

Posted in Uncategorized | Tagged , | Leave a comment

Martingales – Problem (30/365)

Let and be two martingales, . Show that Proceed as follows.

Posted in Uncategorized | Tagged , | Leave a comment

Martingales – Problem (29/365)

Show that every martingale has uncorrelated increments: if then Proceed as follows. As a quick note as to why ,

Posted in Uncategorized | Tagged , | Leave a comment

Martingales – Problem (28/365)

Let the random variables satisfy . Show that the sequence with and where are given functions, is a martingale. Once again, we let the sequence of decompositions be . Then is -measurable because (1) is -measurable, (2) is -measurable, and … Continue reading

Posted in Uncategorized | Tagged , | Leave a comment

Martingales – Problem (27/365)

I am still unable to follow the martingales based proof for the Ballot Theorem, so I’ll work out a few more problems first. A problem asks the following. Let be a sequence of decompositions with , and let be -measurable … Continue reading

Posted in Uncategorized | Tagged , | Leave a comment