Friday, February 20, 2015

mean ^ cond ^ unconditional expectation

Greg Lawler's notes point out that CE is a random variable, and we frequently take UE and variance of a CE. The tower property covers the expectation of CE...

Trivial example: 2 dice rolled. Guess the sum with one revealed.

The CE depends on the one revealed. The revealed value is a RV, so the CE is a "dependent RV". In contrast, the UE (unconditional exp) is determined by the underlying _distribution_, the source of randomness modeled by a noisegen. This noisegen is unknown and uncharacterized, but has time-invariant, "deterministic" properties, i.e. each run is the same noisegen, unmodified. Example - the dice are all the same. Therefore the UE value is deterministic, with zero randomness. The variance of UE is 0.

Now we can take another look at sample mean, a statistical rather than probabilistic concept. Since the sample is a random sample, the mean is a RV, just as the CE is. Variance of sample mean > 0 i.e. if we take another sample the mean may change.

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