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I have been looking for complex Gaussian random variables and Mathematica, however, I haven't found anything on this topic.

Could someone, please, let me know if it is possible to find for example the expectation of a complex Gaussian random variable with Mathematica?

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    $\begingroup$ What have you tried? It's unclear what you're asking: the usual way of specifying such a distribution would have the expectation as input. $\endgroup$ – John Doty Apr 15 '18 at 16:06
  • $\begingroup$ You can create a random variable using RandomVariate[] and you can find the expectation using Expectation[]... $\endgroup$ – Ulrich Neumann Apr 15 '18 at 16:50
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Just use two-dimesional Gaussian random variables and convert them to complex ones. E.g., you may use the following to generate random samples from the "complex" normal distribution:

μ = {0, 0};
Σ = IdentityMatrix[2];
n = 10;
rand = RandomVariate[MultinormalDistribution[μ, Σ], {n}].{1., 1. I}

You can slso use ReIm to convert from complex numbers back to points in the Euclidean plane.

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  • $\begingroup$ That's generation of random variates, but it doesn't address the original question, which is finding the expectation value of something. $\endgroup$ – John Doty Apr 15 '18 at 16:18
  • $\begingroup$ Yeah, I know. It's more a comment with code and a way to make the OP think about the conversion between points in the plane and complex numbers... $\endgroup$ – Henrik Schumacher Apr 15 '18 at 16:20
  • $\begingroup$ @HenrikSchumacher, please, have a look at this question: mathematica.stackexchange.com/questions/171186/…. $\endgroup$ – Felipe Augusto de Figueiredo Apr 15 '18 at 16:27

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