Suppose I have two independent random variables, $X \sim N(\mu_1,\sigma_1^2)$ and $Y \sim N(\mu_2,\sigma_2^2)$ with $\mu_1,\mu_2 > 0$.

How can I compute/estimate

$$ \mathbb{E}\left[\left\lvert \frac{X}{Y} - \frac{\mu_1}{\mu_2}\right\rvert\right] $$

  • $\begingroup$ Is this for homework? $\endgroup$
    – AdamO
    Jan 23 at 15:51
  • 1
    $\begingroup$ @AdamO Unfortunately, I doubt it has such a simple answer that it could be considered as homework $\endgroup$ Jan 23 at 15:53
  • $\begingroup$ Tsk tsk! Is it the simplicity of the answer that determines whether it is homework? Or that it was assigned to you by a teacher/professor in an academic setting? For the latter, out of respect to the teacher and to you, the right "answer" is usually tips and tricks rather than a full treatment - a better if not complete answer, if you will. $\endgroup$
    – AdamO
    Jan 23 at 15:58
  • 3
    $\begingroup$ @Algebro1000 Given its difficulty, you should at least tell us the source of this problem or what motivates you to consider this problem. In addition, "How can I compute/estimate xxx?" is a too vague way to ask a question like this -- to begin with, how to "compute" it is a pure math problem while how to "estimate" it is a statistical problem (and when you try to estimate it, are $\mu_1$ and $\mu_2$ known or unknown?) $\endgroup$
    – Zhanxiong
    Jan 23 at 16:19