Covariance function and mean of process $M(t)=\sqrt{t}Z, \ t\ge0$. $Z\sim N(0,1)$

Question

Let $Z\sim N(0,1)$ and $(M(t),t\ge0)$ be a process with

$$M(t)=\sqrt{t}Z, \ t\ge0$$

I want to determine the mean and covariance function of the process.

$\mu(t)=E[M(t)]=E[\sqrt t Z]=\sqrt t E[Z]=0$

$Cov(M(t),M(s))=Cov(\sqrt t Z,\sqrt s Z)=\sqrt t \sqrt s \ Cov(Z,Z)=\sqrt t \sqrt s \ Var(Z)$

Is that correct?

Answer 1

Yes, it is correct. One can simplify a little bit the expression of the covariance by noticing that the variance of $Z$ is one.