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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

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

Algebro1000
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  • Is this for homework? – AdamO Jan 23 '24 at 15:51
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    @AdamO Unfortunately, I doubt it has such a simple answer that it could be considered as homework – Algebro1000 Jan 23 '24 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. – AdamO Jan 23 '24 at 15:58
  • Similar Qs (not duplicate) https://stats.stackexchange.com/questions/422489/empirical-versus-theoretical-convergence-of-ratio-of-normal-distributions, https://stats.stackexchange.com/questions/28233/distribution-of-ratio-x-y-where-x-is-normal-y-is-half-normal, https://stats.stackexchange.com/questions/178081/distribution-of-ratio-of-2-points-drawn-from-normal-distribution, https://stats.stackexchange.com/questions/202179/what-is-the-distribution-of-the-ratio-of-two-normals and search ... – kjetil b halvorsen Jan 23 '24 at 16:09
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    @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?) – Zhanxiong Jan 23 '24 at 16:19

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