# what is the expectation value of a norm of a random variable from the standard normal distribution?

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

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

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