Complex and Real Zero Mean Gaussian Random Variables

What you would get when you multiply a complex zero mean gaussian random variable with real zero gaussian random variable and what is their expected value?

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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....