I have three complex random variable $x,y,w$.
The pdf of $x$ is ${f_x} \sim CN({\mu _x},\sigma _x^2)$ with mean ${\mu _x}$ and variance $\sigma _x^2$.
The pdf of $y$ is ${f_y} \sim CN({\mu _y},\sigma _y^2)$ with mean ${\mu _y}$ and variance $\sigma _y^2$.
The random variable $w$ is a mixture of $M$ Gaussian component with corresponding mean ${{\mu _{w,i}}}$ and variance ${\sigma _{w,i}^2}$ where,
${f_w} \sim \sum\limits_{i = 1}^M {{\varepsilon _i}CN({\mu _{w,i}},\sigma _{w,i}^2)} $
where $\sum\limits_{i = 1}^M {{\varepsilon _i} = 1} $
Now, for a constant $a,b,c\in R$
$z=ax+by+cw$,
What will be the pdf of $f_z$? How should I express the mean ${\mu _z}$ and variance $\sigma _z^2$? Thank you.