Complex random variable

A complex random variable, like usual complex number has two parts: real and imaginary. The real and imaginary terms are constituted by two real random variables. For example, X = U+jV is a complex random variable where U and V are real random variables. A complex random variable is similar to a joint distribution of two random variables. This setting of complex random variable is important, because it helps in modeling a lot of naturally occurring problems.

Cumulative distribution function of complex random variable X is given by:

F_X(x) = F_UV(u,v) = P(U < u,V < v).

Similarly the probability density function of complex random variable X is given by:

f_X(x) = f_UV(u,v) = ${\frac {\partial }{\partial u}}$ ${\frac {\partial }{\partial v}}$ F_UV(u,v)

A usual probability distribution function of random variable is characterized by the moment functions. Similarly the complex random variable is also characterized by moment functions. The first order moment or the mean of complex random variable X is given by:

$\mu _{x}$ = E{X} = E{U}+jE{V} = $\mu _{u}$ + j$\mu _{v}$

The second order moment of a distribution is its variance. In this case, there are two possible second order moments for the complex random variable. The first type of second order moment has auto correlation content of U and V in it, which is given by:

E{xx^*} = E{(U +j V)(U - jV)} = E{U²+V²} = E{U²} + E{V²}

The above moment is called variance of a complex random variable.

The second type of second order moment of complex random variable X has cross correlation of U and V in it, which is given by:

E{xx} = E{(U +j V)(U + jV)} = E{U²+V²-2jUV} = E{U²} - E{V²} + 2jE{UV}

The above moment is called the complementary variance of a complex random variable. Both second order moments are necessary to completely characterize the complex random variable X.