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Find the sum of Gaussian distribution and Nakagami distribution.

Posted 2 months ago
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I am stuck in a problem in which I need to have knowledge about the sum of Nakagami distribution and Gaussian Distribution. Can anyone enlighten me on what should be the resultant distribution? Is it lies in Gamma distribution?

3 Replies
Posted 2 months ago

What knowledge? Please give more details and show whatever you've tried. For example, are you looking for the pdf, cdf, or mean of the sum of two independent random variables with one having a Nakagami distribution and the other having a Gaussian distribution? And...is this a statistics question or a question as to how to implement this in Mathematica?

Posted 2 months ago

Thank you for your kind reply.

I am looking for the PDF of the sum of two independent random variables with one having a Nakagani distribution and the other having a Gaussian distribution. I wish to have the statistical knowledge on it.

Posted 2 months ago

It would be nice if the following worked:

dist = TransformedDistribution[x1 + x2, {x1 \[Distributed] NormalDistribution[\[Mu], \[Sigma]], 
   x2 \[Distributed] NakagamiDistribution[m, \[CapitalOmega]]}, 
  Assumptions -> {m >= 1/2, \[CapitalOmega] > 0, \[Sigma] > 0}]
PDF[dist, z]

But at least for Mathematica 12.0, it doesn't. That means doing this the old-fashioned way by integrating over the joint distribution of the two random variables:

Integrate[(2 E^(-((m x1^2)/\[CapitalOmega])) x1^(-1 + 2 m) (m/\[CapitalOmega])^m)/Gamma[m]
  E^(-((z - x1 - \[Mu])^2/(2 \[Sigma]^2)))/(Sqrt[2 \[Pi]] \[Sigma]), {x1, 0, \[Infinity]}, 
  Assumptions -> {m >= 1/2, \[CapitalOmega] > 0, \[Sigma] > 0}]

The result for the pdf is

Sum of normal and Nakagami random variables

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