# A list of tricks for calculating expectations?

Does anyone know of a good resource listing known tricks (with examples?) for calculating closed form expressions from messy expectations? (e.g., moment generating function, law of iterated expectations, change of measure, etc.)

In a different setting, I've found Summary of Rules for Identifying ARIMA Models tremendously helpful. I was hoping a list of rules-of-thumb like this would also exist for calculating expectations...right? Unfortunately, I'm not finding anything.

• calculating expectations almost always involves integration. Unfortunately there are no universal simple rules how to calculate integrals. Apr 21, 2011 at 6:54
• @mpiktas I believe anyone who has taught a freshman calculus course might consider disagreeing a little :-).
– whuber
Apr 23, 2011 at 22:07
• @whuber, the key word here is universal. There are simple rules which can help a lot for a large class of integrals. I just wanted to make a point, that there are no similar rules like rules for differentiation. Rereading the question, I see that I missed the point :) Apr 26, 2011 at 14:17

If you are just looking to simplify an expression involving expectations, Economists’ Mathematical Manual has a nice concise list of identities. You can find a copy online here.

• Definitely a nice reference. Didn't find what I was looking for...but I'll hold onto this. I'm sure it'll come in handy at some point :) Apr 27, 2011 at 14:22
• Link is broken (at least when and how I try it). Oct 28, 2018 at 14:35

Let's build it!

You mentioned:

1 moment generating function

2 law of iterated expectations

3 change of measure

4 Decompose random variable as a sum. Usually the sum of indicators of something.

5 Build a reccurence relation for E(X) (or a set of linear equations). Useful in Markov Chains.

6 Stopping time theorem for martingales: $E(X_{T})=E(X_{1})$

6b Wald identity. $E(S_{T})=E(X_{1})E(T)$

7 Kolmogorov forward/backward equation

8 Crofton's method

9 General idea: Symmetry. Especially for "$n$ points are chosen uniformly on the $[0;1]$" problems.

• Right! This is kind of the idea. Maybe should put up a wiki toward which everyone can contribute. (My next project once I'm done with my dissertation :>) Was never aware of Crofton's method btw. Pretty cool! Apr 27, 2011 at 14:21