How to prove $n \geq \frac{(x_1+....+x_n)^2}{(x_1^2+.....+x_n^2)}$?

Asked Jan 19 '23 at 19:37

Active Jan 19 '23 at 20:21

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Currently I try to solve an exercise in which I need to prove that the variance of estimator a is smaller or equal to the estimator. I simplified the equation to the point down below.

$ n \geq \frac{(x_1+....+x_n)^2}{(x_1^2+.....+x_n^2)}$

I am stuck with the form above. I see that this statement in true by plugging numbers in it, however I find it hard to proof it.

Instead of only give the solution, would it be possible to get feedback on the approach of finding such a proof?

edited Jan 19 '23 at 20:21

Sebastiano

7,649

asked Jan 19 '23 at 19:37

Tim

1

Cauchy-Schwarz inequality. – Jyrki Lahtonen Jan 19 '23 at 19:40
1

See here or here. – Jyrki Lahtonen Jan 19 '23 at 19:43
2

This may be an even better duplicate target. – Jyrki Lahtonen Jan 19 '23 at 19:45

How to prove $n \geq \frac{(x_1+....+x_n)^2}{(x_1^2+.....+x_n^2)}$?

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