January 05, 2012

[citation needed]; the difficulty of finding things

Typing in “tower property” in Google, I find that the first result is the ever ubiquitous Wikipedia (whose mastery of SEO means it turns up, with an occasional irrelevant article, on whatever subject you could care to name) article on Expected Value, in this case. Actually typing in “tower property” returns the article on “law of total expectation” which is apparently the one of its myriad names that Wikipedia has decided is most common. Looking at the other results on Google, even adding a helpful “statistics”, I find that “tower property” doesn’t appear to return anything else relevant. In fact, the only other place I can find it called “tower property” is in my notes :)

For nameless results, I find my best bet is simply to type in the result itself. For example, that E[XY]=E[YE[X|Y]] is proven at the end of this pdf document, which is likely lecture notes. If something has a lot of roots or powers, this is somewhat less applicable.

As of yet, I’ve not been able to find anything on what my notes refer to as “Fisher’s theorem”. It’s a theorem named after a famous mathematician who had many theorems named after him (some with others), so we’re already off the a bad start trying to find it. The theorem reads:

Let X_i \sim N(\mu, \sigma^2) be indepedent random variables. Define \overline{X}=\sum_{i=1}^n X_i and s^2=\frac{1}{n-1}\sum_{i=1}^n (\overline{X}_n-X_i)^2. Then:
*\overline{X} \sim N(\mu,\frac{\sigma^2}{n})
*\overline{X} and s^2 are independent.
*\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}
*\frac{\overline{X}-\mu}{\sqrt{s^2/n}} \sim t_{n-1}

It looks like it has something to do with sample mean and variance, but I’m only taking the first module on this topic, so what its use is I can’t say.


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