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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1$\begingroup$ Welcome to Cross Validated! Are the components independent? $\endgroup$– DaveApr 26, 2022 at 14:53
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$\begingroup$ Thank you, yes they are independent. $\endgroup$– RazApr 26, 2022 at 15:12
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1$\begingroup$ 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 $p$. $\endgroup$– mloftonApr 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.
$$ \mathbb E\bigg[\vert\vert\zeta\vert\vert^2\bigg] = \mathbb E\bigg[ \sum_{i=1}^k \zeta_i^2 \bigg] = \sum_{i=1}^k\mathbb E\bigg[ \zeta_i^2 \bigg] = \sum_{i=1}^k 1 = k $$
