$\begin{vmatrix}~E\Big[\sum \limits_{i=1}^{n}X_i\Big]~\end{vmatrix} \le \sum \limits_{i=1}^{n}\begin{vmatrix}E[X_i]\end{vmatrix}$

Asked Aug 13 '20 at 13:55

Active Aug 13 '20 at 13:55

Viewed 27 times

What's the name of this inequality:

$$\begin{vmatrix}\sum \limits_{i=1}^{n}X_i\end{vmatrix} \le \sum \limits_{i=1}^{n}\begin{vmatrix}X_i\end{vmatrix}$$

$$\begin{vmatrix}~E\Big[\sum \limits_{i=1}^{n}X_i\Big]~\end{vmatrix} \le \sum \limits_{i=1}^{n}\begin{vmatrix}E[X_i]\end{vmatrix}$$

asked Aug 13 '20 at 13:55

pico

1

Triangle inequality? – kimchi lover Aug 13 '20 at 14:04
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you even have $$\left|\mathbb E\left[\sum_{i=1}^nX_i\right]\right|\leq \sum_{i=1}^n\mathbb E[|X_i|]$$ – Surb Aug 13 '20 at 14:04
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The second inequality also involves the linearity of the esperance. – TheSilverDoe Aug 13 '20 at 14:08

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