Let $W=\{W_t:t\geq0\}$ be a Brownian motion. Find $\operatorname{Var}(W_1^3)$

Question

sorry for my newbie question. I'm learning stochastic process on my own and I got confused about a lot of concepts and notations. I need your help with this small exercise:

Let $W=\{W_t:t\geq0\}$ be a Brownian motion. Find $\operatorname{Var}(W_1^3)$

Please could you give me a little bit of explanation on how to approach a problem like this?

Answer 1

Note $W_1=(W_1-0)=(W_1-W_0)\sim N(0,1-0)=N(0,1)$. Then, $$ \operatorname{Var}(W_1^3)=E(W_1^6)-[E(W_1^3)]^2=15-0^2=15. $$ In a similar manner, you can show that $\operatorname{Var}(W_t^3)=15t^3$.