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

Solution 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.