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How do I prove this? (summation problem) 
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#1
Mar2414, 07:58 PM

P: 56

$$\sum_{i=1}^{n} x_i^2 > \frac{1}{n^2}(\sum_{i=1}^{n} x_i)^2$$
Note: each x_i is any observation (or statistic) it can be any real number and need not be constrained in anyway whatsoever, though you can take n > 1 and integer (i.e. there is at least two observations and the number of observations is discrete). I'm not sure if this true or not, but part of my analysis to a particular problem assumed this was true, and I'm trying to prove it is indeed true (it seems to be case for any examples I come up with). So far I came up with, $$n^2 \sum_{i=1}^{n} x_i^2 > \sum_{i=1}^{n} x_i^2 + 2\sum_{i \neq j, i > j} x_ix_j$$ $$(n^2  1)\sum_{i=1}^{n}x_i^2 > 2\sum_{i \neq j,\: i > j} x_ix_j$$ and I'm not sure how to proceed from there. 


#3
Mar2414, 08:09 PM

P: 56




#4
Mar2414, 08:09 PM

Mentor
P: 18,334

How do I prove this? (summation problem)



#5
Mar2414, 08:17 PM

P: 56

$$(\sum_{i=1}^{n} x_i \times 1)^2 \le (\sum_{i=1}^{n}x_i^2) (\sum_{i=1}^{n}1) = n\sum_{i=1}^{n}x_i^2 < n^2 \sum_{i=1}^{n}x_i^2$$ Which was what I wanted. Thanks! 


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