p.d.f of the absolute value of a Gaussian random variable of non-zero mean

Question

For a complex random variable (r.v), if real and imaginary parts are i.i.d with Normal distribution, the absolute value of the r.v. follow Rayleigh distribution. However, what if the real and imaginary parts are Gaussian distributed with arbitrary non-zero mean?

Answer 1

Sorry for my previous answer; I was wrong. As pointed out by @JonasDahlbæk:

• the $L_2$-norm of a vector of centred normal r.v. with unit variance is chi-distributed;
• if the r.v. have different means (but still unit variance), then the norm is noncentral chi-distributed;
• if the r.v. have different means and variances, then based on this paper, I think the distribution of the norm cannot be formulated in terms of elementary functions. There is a also a related question on Mathoverflow, with a selected answer pointing to the book "Quadratic Forms in Random Variables" by Mathai and Provost.

I do not have time to summarise the main points of this paper right now, but I intend to do so in the near future. In the meantime, perhaps the reference can help you, or perhaps someone else can provide a better answer.