Expectation of a squared random variable and of its absolute value

Question

Is it true that if $\mathbb{E}[X^2]<\infty$ then $\mathbb{E}[|X|]<\infty$? If so, why?

Answer 1

The most direct way to see this, without referring to a single theorem, is to consider the set $E=\{|X| \leq 1\}$. Then if $1_E$ is the indicator function of $E$,

$$\mathbb{E}[|X|] = \mathbb{E}[|X|\cdot 1_E] + \mathbb{E}[|X| \cdot 1_{E^c}] \leq \mathbb{E}[1 \cdot1_E] + \mathbb{E}[X^2 \cdot 1_{E^c}] \leq1+\mathbb{E}[X^2] < \infty.$$

The more clever arguments using Jenson's or Holder's inequalities provide a much sharper bound, however.

Answer 2

Yes, it is true. Apply Jensen inequality to the convex function $x \mapsto x^2$ or Cauchy-Schwarz inequality $\Bbb E[|XY|] \leq \sqrt{\Bbb E[|X|^2] \Bbb E[|Y|^2]}$ with $Y = 1$.

Answer 3

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$$\imp\qquad\color{#0000ff}{\large% \verts{\vphantom{\LARGE A}\angles{x}}\ \leq\ \angles{x^{2}}^{1/2}} $$