The sum of squared zero-mean Gaussian random variables with different variances can be modeled by a generalized Chi-square distribution. However, I am looking for the PDF of the sum of squared Gaussian random variables with different means and variances. Thank you in advance!
I do not know the exact answer, but you could find something related (or the answer) in the book Quadratic Forms in Random Variables by Mathai & Provost and on the paper On the Generalized Noncentral Chi-Squared Distribution Induced by an Elliptical Gamma Law by Markos Koutras.
Thank you by your answer my friend! I'll read the book. At this moment, I'm trying to do something with the moment generating function of the random variable.
The sum of these two random variables can be modeled as a noncentral chi-square distribution. The noncentral chi-square distribution is a generalization of the chi-square distribution, which models the sum of Gaussian random variables with different means and unit variances. I don't know if, in your case, the random variables have unit variance, but in this case, you can normalize them.
Rafael Chaves thank you so much by your answer. In fact, I need to obtain the PDF of a linear combination of the sum of independent squared zero-mean Gaussian random variables. These Gaussian random variables have unit variances. The problem is that there are factors that multiply them and they change the variances of these random variables. As example a(x_1 + x_2) + b(x_3 + x_4 + x_5) + cx_6 where a, b and c are constants and x_i are squared gaussian random variables with no-zero mean and unit variance.
Henry Carvajal your problem can be seen as a sum of 3 chi-squared random variables with different degrees of freedom, like aX^2(2) + bX^2(3) + cX^2(1). Unfortunately, a closed expression for this problem is not know, but you can approximate it by using the property of characteristic functions of chi-square random variables. I believe this paper can be useful for you " On the efficient calculation of a linear combination of chi-square random variables with an application in counting string vacua"
Link: https://iopscience.iop.org/article/10.1088/1751-8113/46/50/505202/pdf
Dear Rafael Chaves thank you so much by your help and your recommendation. I will check the paper.
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