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 $Z$?
Probability density function of a complex-valued random variable
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probability
statistics
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0This would be the definition of the cumulative probability distribution function of any joint random variable. However, this is a particular joint variable because $z=u+iv$. Hence I was considering the transformation of random variable $UV$ to include this particularity, but I'm sure this is wrong as I didn't find anything like it in the books I've read. – 2012-07-20
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Complex number is treated as 2-d real vector here.