#Sigma math problem how to#

Probability, sigma algebras, and random variables Conditional expectation of a random variable with respect to a sigma algebra.Expected value of a random variable and conditional probability.

Probability, sigma algebras, and random variables.

I would like to thank Everule and rivalq for suggesting me to write this blog, and them and meme and adamant for proofreading and discussing the content to ensure completeness and clarity.īONUS: Here's an interesting paper that uses martingales to solve stuff. The concept of 'minimal' sets is used for a lot of the mental modelling in the blog, so maybe you'd still want to read the relevant parts of the blog where it is introduced. People already familiar with the initial concepts can skip to the interesting sections, but I would still recommend reading the explanations as a whole in case your understanding is a bit rusty.

#Sigma math problem how to#

Also, note that sometimes the explanations will have more material than is strictly necessary, but the idea is to have a more informed intuition rather than presenting a terse-yet-complete exposition that leaves people without any idea about how to play around with the setup. I will do this because people tend to get lost in the measure-theoretic aspects of the rigor that is needed and skip on the intuition that they should be developing instead. Note that in the spirit of clarity, I will only be dealing with finite sets and finite time processes (unless absolutely necessary, and in such cases, the interpretation should be obvious). So I hope the following intuitive introduction helps people develop a deeper understanding. I think a lot of people don't get the intuition behind these topics and why they're so useful. It turned out that these problems were solvable using something called martingales which are random processes with nice invariants and a ton of properties. Note that here the time to completion is a random variable, and in probability theory, such random variables are called 'stopping times' for obvious reasons. Recently someone asked me to explain how to solve a couple of problems which went like this: 'Find the expected time before XYZ happens'.