# Ratio of Normal Distributions [duplicate]

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

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

• Is this for homework? Jan 23 at 15:51
• @AdamO Unfortunately, I doubt it has such a simple answer that it could be considered as homework Jan 23 at 15:53
• 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. Jan 23 at 15:58
• @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?) Jan 23 at 16:19