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Tag Archives: probability
Probability Foundations – Problem (38/365)
Let be a finitely additive measure on an algebra , and let be pairwise disjoint and satisfy . Then show that .
Probability Foundations – Problem (37/365)
A problem similar to the previous post. Let be a countable set and a collection of all its subsets. Put if is finite and if is inifinite. Show that the set function is finitely additive but not countable additive. To … Continue reading
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
Expectation Maximization – Link (35/365)
Dan Piponi has written up a simple to follow derivation of the ExpectationMaximization 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
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
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
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
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
Martingales – Problem (30/365)
Let and be two martingales, . Show that Proceed as follows.
Martingales – Problem (29/365)
Show that every martingale has uncorrelated increments: if then Proceed as follows. As a quick note as to why ,