Natural definition for variance?

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Homework Statement



I'm trying to find a natural definition for V[X|Y], but I'm not sure what a natural definition is. Any help?
 
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This already has a definition. It's the second moment of X where you integrate against the pdf condition on Y. I guess I'm confused about your question.
 
Actually I'm assuming that you're asking about V[X|Y = y] which is just a deterministic value. The way you wrote it V[X|Y] would actually be a random variable itself. Look at the second bullet here:

http://en.wikipedia.org/wiki/Conditional_expectation#Discussion
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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