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I require the proof for the following

$$\sum_{i = 1}^N x_i^2 \ge N \mu ^2 $$

where $x_i \in \mathbb R$ and

$$\mu = \frac{1}{N} \sum_{i = 1}^N x_i$$

I can visually see how this is true (I imagine rectangles and squares), and would like to know if there's a common name for the result. If not, what's an easy way to show a proof for this? Thanks!

B2VSi
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  • in what context do you require this proof? what have you tried? how about doing this for small $N$ to build intuition? e.g. for $N=1$, does this always hold? what about for $N=2$? – gt6989b Apr 24 '23 at 16:49
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    I need it to prove a result I have (that uses this). I know this result is true, I just want some easy way for me to explain how I am able to use it. A name works best, but if there's none, a simple proof that's easy to follow would be nice. I don't want to take time/space explaining something that's not as important/relevant to my study. – B2VSi Apr 24 '23 at 17:00

2 Answers2

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Suppose the random variable $X$ takes $x_i,~i=1,2,...,N$ with uniform probabilities, which is:

$$P(X=x_i)=\frac{1}N,~~~~~~~~i=1,2,...,N$$

So we have $Var(X)=E(X^2)-E^2(X)\ge0$, which gives:

$$E(X^2)=\frac{1}{N}\sum_{i = 1}^N x_i^2 \ge E^2(X)= \left(\frac{1}{N} \sum_{i = 1}^N x_i\right)^2=\mu^2 $$

Done.

MathFail
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The inequality $$\frac{\sum_{i=1}^N x_i^2}{N} \ge \left( \frac{\sum_{i=1}^N x_i}{N}\right)^2$$ may be called Cauchy–Schwarz inequality and can be proved by

NN2
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