5
$\begingroup$

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.

$\endgroup$
3
  • 3
    $\begingroup$ calculating expectations almost always involves integration. Unfortunately there are no universal simple rules how to calculate integrals. $\endgroup$
    – mpiktas
    Apr 21, 2011 at 6:54
  • 1
    $\begingroup$ @mpiktas I believe anyone who has taught a freshman calculus course might consider disagreeing a little :-). $\endgroup$
    – whuber
    Apr 23, 2011 at 22:07
  • $\begingroup$ @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 :) $\endgroup$
    – mpiktas
    Apr 26, 2011 at 14:17

2 Answers 2

2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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 :) $\endgroup$
    – lowndrul
    Apr 27, 2011 at 14:22
  • $\begingroup$ Link is broken (at least when and how I try it). $\endgroup$
    – Nick Cox
    Oct 28, 2018 at 14:35
3
$\begingroup$

Let's build it!

You mentioned:

1 moment generating function

2 law of iterated expectations

3 change of measure

Adding:

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.

$\endgroup$
1
  • $\begingroup$ 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! $\endgroup$
    – lowndrul
    Apr 27, 2011 at 14:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.