Show that the process $X=(W_{\sqrt{t}}I_{(1,2)}(t))_{t \ge 0} \in \mathcal{L}_3^2$. ($W$- Wiener) Additionally calculate, for $t,s \in [1,2]$, $EX_t$ and $Cov(X_t,X_s)$
I have no idea how to start with it. Please help me :(
Is it just : $EX_t=EW_{\sqrt{t}}=0$ ? And for $s \le t$ $Cov(X_t,X_s)=Cov(W_{\sqrt{t}},W_{\sqrt{s}})=\sqrt{s}$