Expectation of norm squared for vector of i.i.d. normal RVs

[OhMyMarkov]

Hello everyone! :D

Suppose $v$ is a vector of i.i.d. normal RV's with zero mean and standard deviation $\sigma$. Is the following true:

(1) $E[||v||^2]=\sigma ^2$
(2) $E[\sum _i v_i] = 0$

Thank you for your help!

 

[CaptainBlack]

Hello everyone! :D

Suppose $v$ is a vector of i.i.d. normal RV's with zero mean and standard deviation $\sigma$. Is the following true:

(1) $E[||v||^2]=\sigma ^2$
(2) $E[\sum _i v_i] = 0$

Thank you for your help!

\[E\left(||v||^2\right)=E\left(\sum_i v_i^2 \right)=\sum_i E(v_i^2)=n\sigma^2\]

Now do the same process of using the expectation of a sum is the sum of the expectations on the second.

CB