[1] Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts.
Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.
Other concepts are unique to complex random variables.
Applications of complex random variables are found in digital signal processing,[2] quadrature amplitude modulation and information theory.
Consider a random variable that may take only the three complex values
The expectation of this random variable may be simply calculated:
Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set
The density function is shown as the yellow disk and dark blue base in the following figure.
Complex Gaussian random variables are often encountered in applications.
They are a straightforward generalization of real Gaussian random variables.
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form
of a complex random variables via the joint distribution of their real and imaginary parts: The probability density function of a complex random variable is defined as
, i.e. the value of the density function at a point
is defined to be equal to the value of the joint density of the real and imaginary parts of the random variable evaluated at the point
As in the real case the density function may not exist.
The expectation of a complex random variable is defined based on the definition of the expectation of a real random variable:[3]: p. 112 Note that the expectation of a complex random variable does not exist if
is linear in the sense that for any complex coefficients
It is equal to the sum of the variances of the real and imaginary part of the complex random variable: The variance of a linear combination of complex random variables may be calculated using the following formula: The pseudo-variance is a special case of the pseudo-covariance and is defined in terms of ordinary complex squares, given by: Unlike the variance of
For a general complex random variable, the pair
Its elements equal: Conversely: The covariance between two complex random variables
is defined as[3]: 119 Notice the complex conjugation of the second factor in the definition.
In contrast to real random variables, we also define a pseudo-covariance (also called complementary variance): The second order statistics are fully characterized by the covariance and the pseudo-covariance.
The covariance has the following properties: Circular symmetry of complex random variables is a common assumption used in the field of wireless communication.
A typical example of a circular symmetric complex random variable is the complex Gaussian random variable with zero mean and zero pseudo-covariance matrix.
By definition, a circularly symmetric complex random variable has
Thus the expectation of a circularly symmetric complex random variable can only be either zero or undefined.
Thus the pseudo-variance of a circularly symmetric complex random variable can only be zero.
This means that a complex random variable is proper if, and only if: Theorem — Every circularly symmetric complex random variable with finite variance is proper.
For a proper complex random variable, the covariance matrix of the pair
has the following simple form: I.e.: The Cauchy-Schwarz inequality for complex random variables, which can be derived using the Triangle inequality and Hölder's inequality, is The characteristic function of a complex random variable is a function