How does the law of iterated expectation work?
The Law of Iterated Expectation states that the expected value of a random variable is equal to the sum of the expected values of that random variable conditioned on a second random variable.
How is the expected outcome of an event calculated?
Intuitively speaking, the law states that the expected outcome of an event can be calculated using casework on the possible outcomes of an event it depends on; for instance, if the probability of rain tomorrow depends on the probability of rain today, and all of the following are known:
Which is a simple proof of the law of iterated expectations?
$\\begingroup$A simple proof is provided in the lecture note (math.arizona.edu/~tgk/464_07/cond_exp.pdf), see theorem 8 on page 4.$\\endgroup$– WenxuMay 31 at 3:01 Add a comment | 3 Answers 3 ActiveOldestVotes 46 $\\begingroup$ INFORMAL TREATMENT
Where does the property of conditional expectation come from?
They are of course right: this important and very intuitive property of conditional expectation derives essentially directly (and almost immediately) from its definition -the only problem is, I suspect that this definition is not usually taught, or at least not highlighted, outside probability or measure theoretic circles.
Which is the decomposition of variance in iterated expectations?
Decomposition of variance (Wooldridge, p. 31) • Proof that var(y) = var. x[E(y|x)]+E. x[var(y|x)] (i.e., the variance of y decomposes into the variance of the conditional mean plus the expected variance around the conditional mean).
Do you know the law of iterated expectations for Lie?
There are tons of questions related to the LIE, but all the ones I’ve seen do not help in my case, including this one and this one. I know that by Law of Iterated Expectations (LIE), E(xi | Ai) = 0 implies E(xi) = 0 and E(xiAi) = 0.
When do conditional expectations need to be zero?
This does not require any independence, but it does require that your conditional expectations are zero for all values of the auxiliary variables. To show this mathematically, I will start with your first case (single auxiliary variable), then build to the more general case.