Currently, I am coming to grips with the law of total variance.
In words, the variance of X equals the variance of the expected value of X, given Y, plus the expected value of the variance of X, given Y.
In symbols, Var(X) = Var(E(X|Y)) + E(Var(X|Y).
By reading the verbal definition, one can see that the logic is convoluted. Given two probability distributions, it can be tricky to see how to apply the law. Once the law is applied, there are additional tricky steps of logic involving the independence of variables.
From the SOA/CAS sample problems:
A motorist makes 3 driving errors, each independently resulting in an accident with probability 0.25. Each accident results in a loss that is exponentially distributed with mean 0.80. Losses are mutually independent and independent of the number of accidents. The motorists insurer reimburses 70% of each loss. Find the variance of the total unreimbursed loss.
The above item illustrates the technical problem that I have been having when using Spaced Repetition Software to schedule repetitions of complex material.
When I encounter a difficult item, I spend time exploring the given solution, and alternate solutions. Eventually, I move on to another item. Most likely, I am not entirely comfortable with the material, and I would like to see the same material every day for a while, to approach it with different solutions. The way that the defaults on SRS software are set up, as soon as you start rating an item anything other than the most difficult setting, the item starts getting pushed off way into the future. After a week or two weeks, I have entirely forgotten many of the finer details of the item, and it is almost as if I am starting from scratch. For difficult material, it is beneficial to see the same items every day, or every 2nd or 3rd day.
The solution to this problem is to reset the forgetting index. In Anki, I have now set my forgetting index to 3%. This task is done by downloading the shared “Forgetting Index” plugin. In the “File” menu, select “Download Shared Plugin”, and find the forgetting index plugin.
The result is that I can now look at problems on a nearly daily basis before they start whizzing into the future. Each time I look at the problem with fresh eyes, I observe new things.
In the last week, I finally have my Spaced Repetition Software working to its utmost. I use it to schedule repetitions of difficult exam problems, and of memorization items. These are two fundamentally different tasks, so it is important to set up Anki (the software that I am now using), in two different ways. After spending time every day for a week with the above “unreimbursed loss” problem, I look at it, and think “piece of cake.” Now, when I encounter other law of total variance situations, I have several good comparison problems stashed away in my head.