On probability density functions for complex variables

Complex random variables arise naturally in many settings and their properties are of general interest. Past work on complex variables has mainly focused on their second-order structure, as well as that of their conjugates, whereas the main purpose of this correspondence is to clarify the concept of a density function for a complex random variable, and to discuss its properties. Two different functions play the role that the density of a real univariate random variable holds. Only one of these two functions can be correctly interpreted as a density, but both functions clarify the nature of a complex variable. The role played by the complex conjugate of the variable in this formulation is clarified, and the complex scalar nature of Z is discussed. As the properties of complex random variables are most naturally specified in terms of the complex quantities directly, and given in terms of the distribution of the complex variables rather than formulated in terms of the real and imaginary parts, ensuring that an interpretable complex formulation exists is important