If $a$, $b$, $c$, and $d$ are jointly Gaussian complex random variables then what will be the value of $$\mathbb{E}[abcd]\ ?$$
I found two formulas regarding this and I am confused which one is correct. Is it
$$ \mathbb{E}[abcd] = \mathbb{E}[ab]\mathbb{E}[cd] + \mathbb{E}[ac]\mathbb{E}[bd] + \mathbb{E}[ad]\mathbb{E}[bc] - 2\mathbb{E}[a]\mathbb{E}[b]\mathbb{E}[c]\mathbb{E}[d] $$ or $$ \mathbb{E}[a^* b^* cd] = \mathbb{E}[a^* c]\mathbb{E}[b^* d] + \mathbb{E}[a^* d]\mathbb{E}[b^* c]$$ where $a^*$ is complex conjugate of $a$?
Here are the snapshots of the books where I found those formulas. For the first formula:
And for the second formula:
Edit: I found one more now:
