This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.To say that M is a complex Gaussian Random Variable with distribution N(0,s²), means that its real and imaginary parts are iid and have distribution N(0,s²). Let us assume that G is a real variable also with distribution N(0,s²), then we can consider the variable U=MG it is a complex random variable, with a non-trivial distribution (pdf).
If we assume that M and G are independent, we obtain however that the expectation of a product remains easy to compute: E(U)=E(M).E(G).
Since E(M)=E(Re(M))+iE(Im (M))=0+i0=0, and since E(G)=0, we have that E(U)=0.
Now consider the case that we have random variables M(t1), G(t1),....M(t4), G(t4) with the indicated distributions. When the times are different we assume that these random variables are independent, while when some of them are the same, we will need to analyze that further....