0

suppose I have random variable X and Y with uncertainty $σ_x$ and $σ_y$. Now I want to estimate the uncertainty of $\frac XY$.

when σx is far less than Mean[X] and σy far less than Mean[Y], this problem has an easy resolution. However, I want to get the error bar of $\frac XY$ when σx is not necessary far less than Mean[X] and similarly for $σ_y$.

So this this the abstract model:

suppose$X~\sim~N(\bar x,σ_x^2),Y \sim~N(\bar y,σ_y^2)$ what is the distribution of $\frac XY$? (we know it is Cauchy distribution if $\bar x=\bar y=0$ and $σ_x=σ_y$ by Normal Ratio Distribution with CDF Method)

or what I care most, what is Mean[$\frac XY$] or most probable $\frac XY$?

If I want to set an confidence interval (error bar/uncertainty estimation) of ,saying 75%, for $\frac XY$ distribution, how should I chose? It seems that [ $\frac{\bar x-σ_x}{\bar y+σ_y}$,$\frac{\bar x+σ_x}{\bar y-σ_y}$] is a nature one, what is the confidence for this interval?

If I have $\bar x-σ_x$ and $\bar y-σ_y$ significantly large than $0$, will this problem be simplified?

Community
  • 1
Harry
  • 111
  • MathJax formatting – JKnecht Mar 02 '16 at 08:08
  • by the way, may/how could I just copy similar MathJax formatting expressions from exist post in this site and modify them (which would be convenient)? I 'm newbie to MathJax formatting... @JKnecht – Harry Mar 02 '16 at 08:40
  • See the following paper "ratios of Normal Variables" by G. Marsaglia (Journal of Stistical Software, May 2006, Vol 16, Issue 4. https://www.jstatsoft.org/article/view/.../v16i04.pdf – Jean Marie Mar 02 '16 at 09:42
  • I can't download the article from the link you provide, which redirected to https://www.jstatsoft.org/index instead... @JeanMarie – Harry Mar 02 '16 at 10:27
  • 1
    All right, I understand: I have through my university transparent access to some data basis. You can now download this paper :

    https://filezemp.univ-st-etienne.fr/yr6k8

     – Jean Marie Mar 02 '16 at 10:43
  • One obvious difficulty arises when X and Y are both standard normal. Then the ratio has a Cauchy distribution for which neither mean nor variance exists. Intuitively, there are 'too many' values near 0 in the denominator so that the distribution of the ratio has very heavy tails. Are there some restrictions missing in your statement of the problem? – BruceET Mar 02 '16 at 18:55
  • Ahh...my fault. I take for granted the restriction of $\bar x =\bar y =0$ can be released to $\bar x =\bar y$, which is not true. – Harry Mar 03 '16 at 04:59
  • @BruceET as mentioned in the last sentence, I'm interested in the condition where "If I have $\bar x−σ_x$ and$\bar y−σ_y$ significantly large than 0" – Harry Mar 03 '16 at 05:01
  • Has the download been successful ? – Jean Marie Mar 03 '16 at 06:40
  • @JeanMarie yes. Thank you very much. The article is really useful! I was somehow surprised to find that there is a "modern" article for this "old" "classical" problem which I supposed should be solved hundred years ago and written down in some unknown textbook I hadn't read before... .... – Harry Mar 03 '16 at 07:03
  • @JeanMarie considered that the article might be useful to those dealing data with some problem as me, is there anyway to make that link permanent? – Harry Mar 03 '16 at 07:07
  • Its not possible on this server (it will stay there for 10 days). For present time, people who need it can ask me. – Jean Marie Mar 03 '16 at 07:14

0 Answers0