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Concept# Complex random variable

Summary

In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. 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.
Some concepts of real random variables have a straightforward generalization to complex random variables—e.g., the definition of the mean of a complex random variable. Other concepts are unique to complex random variables.
Applications of complex random variables are found in digital signal processing, quadrature amplitude modulation and information theory.
A complex random variable on the probability space is a function such that both its real part and its imaginary part are real random variables on .
Consider a random variable that may take only the three complex values with probabilities as specified in the table. This is a simple example of a complex random variable.
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 . This random variable is an example of a complex random variable for which the probability density function is defined. The density function is shown as the yellow disk and dark blue base in the following figure.
Complex normal distribution
Complex Gaussian random variables are often encountered in applications. They are a straightforward generalization of real Gaussian random variables. The following plot shows an example of the distribution of such a variable.
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form make no sense. However expressions of the form make sense.

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In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. 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.

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In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.

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