stylistic:tone
Complex random variable

The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form P (Z ≤ 1 + 2 i) {\displaystyle P(Z\leq 1+2i)} make no sense. However expressions of the form P (ℜ (Z) ≤ 1, ℑ (Z) ≤ 3) {\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)} make sense. Therefore, we define the cumulative distribution of a complex random variables via the joint distribution of their real and imaginary parts: F Z (z) = F ℜ (Z), ℑ (Z) (ℜ (z), ℑ (z)) = P (ℜ (Z) ≤ ℜ (z), ℑ (Z) ≤ ℑ (z)). {\displaystyle F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})=P(\Re {(Z)}\leq \Re {(z)},\Im {(Z)}\leq \Im {(z)}).}