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In the middle of an argument that I will present to my students I'll have to prove that $E[X^2]\geq E^2[X]$, but I don't want to use Jensen's inequality to do so. Is there any elementary way to go?

Thanks!

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It follows from $\mathrm{Var}[X] = \mathrm{E}[X^2] - \mathrm{E}^2[X] \geq 0$.

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    $\begingroup$ +1 This is ideal because it relies on the most basic inequality of all: that squares of (real) numbers are non-negative. Even schoolchildren readily grasp that! $\endgroup$
    – whuber
    Apr 27, 2012 at 14:48
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    $\begingroup$ (+1) Indeed, this is precisely the example I keep in my head when thinking about Jensen's inequality so that I don't accidentally get the inequality in the wrong direction. $\endgroup$
    – cardinal
    Apr 27, 2012 at 15:30
  • $\begingroup$ I do exactly the same "mind trick", cardinal! $\endgroup$
    – Zen
    Apr 27, 2012 at 16:50

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