There is a part of question below.
Assume $X_n$ is an i.i.d. gaussian random process with zero mean and variance $\sigma^2$, find its covariance.
About this part, I have a question, it said $X_n$ is a process, but according to his description, zero mean and variance $\sigma^2$, I think $X_n$ should be a variable, that is, $f_X(x)=\frac{1}{ \sigma \sqrt{2 \pi} }e^{-\frac{1}{2} (\frac{x-m_x}{\sigma})^2}$. Because there are both mean and variance in this pdf, but for the process, there is just mean in the pdf.
Gaussian random process: $\mathbf m$ is mean, and $\mathbf C$ is covariance matrix
$$f_\mathbf X(\mathbf x)=\frac{1}{(2\pi )^{\frac{n}{2}}\sqrt{\det \mathbf C}} e^{-\frac{1}{2}(\mathbf x - \mathbf m)^T \mathbf C^{-1}(\mathbf x - \mathbf m)}$$
So I want to ask is it possible that the question is wrong, that is, the $X_n$ is actually a variable, not a process? Or if the $X_n$ is actually a process, then how do I calculate the covariance of Gaussian random process when I know the zero mean and variance $\sigma^2$?