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 $E[|| \zeta_k ||^2]=p$?

  • 1
    $\begingroup$ Welcome to Cross Validated! Are the components independent? $\endgroup$
    – Dave
    Apr 26, 2022 at 14:53
  • $\begingroup$ Thank you, yes they are independent. $\endgroup$
    – Raz
    Apr 26, 2022 at 15:12
  • 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 $p$. $\endgroup$
    – mlofton
    Apr 26, 2022 at 15:16

1 Answer 1


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.

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.