[Math] Probability density function of a complex-valued random variable

Question

I'm trying to understand the concept of complex-valued random variables, but I'm struggling. If you consider two real-valued random variables $U$ and $V$ with values $u$ and $v$ and the joint random variable $UV$ with values $(u,v)$ then under the following transformation of random variable $UV$ to random variable $ZW$
$$z=u+iv$$
$$w=u$$
the probability density function of $ZW$ is (Jacobian determinant =1)
$$p_{ZW}(z,w)=p_{UV}(z-iw,w)$$
and the marginal probability density function $p_{Z}(z)$ is then given by
$$p_{Z}(z)=\int_{-\infty}^{\infty}{p_{UV}(z-iw,w)}dw$$
Is this the probability density function of complex-valued random variable $Z$?

Answer 1

Complex number is treated as 2-d real vector here.