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Let's suppose $\zeta_k \in \mathbb{R}^p$ is a random variable from a normal distribution with mean zero and 1 standard deviation. why $E[|| \zeta_k ||^2]=p$?

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    $\begingroup$ Welcome to Cross Validated! Are the components independent? $\endgroup$
    – Dave
    Commented Apr 26, 2022 at 14:53
  • $\begingroup$ Thank you, yes they are independent. $\endgroup$
    – Raz
    Commented Apr 26, 2022 at 15:12
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    $\begingroup$ The expectation, $E(\eta_k)^2$, is 1 and you can sum them since they are independent and since there are $p$ of them, the result is $p$. $\endgroup$
    – mlofton
    Commented Apr 26, 2022 at 15:16

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Since the components are independent, it means that the squared $L_2$ norm of $\zeta$ has a $\chi^2_k$ distribution, which is the sum of squared independent standard normal variables.

The expected value of a $\chi^2_k$ variable is $k$.

In more generality, we do not even require the independence.

$$ \mathbb E\bigg[\vert\vert\zeta\vert\vert^2\bigg] = \mathbb E\bigg[ \sum_{i=1}^k \zeta_i^2 \bigg] = \sum_{i=1}^k\mathbb E\bigg[ \zeta_i^2 \bigg] = \sum_{i=1}^k 1 = k $$

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