# What is the ratio of uniform and normal distribution?

Let $X$ follow a uniform distribution and $Y$ follow a normal distribution. What can be said about $\frac X Y$? Is there a distribution for it?

I found the ratio of two normals with mean zero is Cauchy.

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For what it's worth, the distribution of $Y/X$ is called the slash distribution. I don't know if the reciprocal has a name or a closed form. –  David J. Harris Mar 27 at 4:31
And the larger class to which both belong seems to be ratio distributions! –  Nick Stauner Mar 27 at 4:33
@DavidJ.Harris Quite so; +1. I've seen the slash used a few times in robustness studies. Maybe $X/Y$ - as an inverted slash - should be called the "backslash distribution". –  Glen_b Mar 27 at 4:46
@rrpp Are you referring to a standard $Uniform(0,1)$, or a general $Uniform(a,b)$? If the latter, then we need to know if $a>0$, $a<0$ etc. –  wolfies Mar 27 at 5:15
thank you all for your answers. @wolfies $X$ is $Uniform(0,1)$ and $Y$ has positive mean –  rrpp Mar 27 at 5:38

Let random variable $X \sim Uniform(a,b)$ with pdf $f(x)$:

where I have assumed $0<a<b$ (this nests the standard $Uniform(0,1)$ case). [ Different results will be obtained if say parameter $a<0$, but the procedure is exactly the same. ]

Further, let $Y \sim N(\mu, \sigma^2)$, and let $W=1/Y$ with pdf $g(w)$:

Then, we seek the pdf of the product $V = X*W$, say $h(v)$, which is given by:

where I am using mathStatica's TransformProduct function to automate the nitty-gritties, and where Erf denotes the Error function: http://reference.wolfram.com/language/ref/Erf.html

All done.

Plots

Here are two plots of the pdf:

• Plot 1: $\mu = 0$, $\sigma = 1$, $b = 3$ ... and ... $a = 0, 1, 2$

• Plot 2: $\mu = {0,\frac12,1}$, $\sigma = 1$, $a=0$, $b = 1$

Monte Carlo check

Here is a quick Monte Carlo check of the Plot 2 case, just to make sure no errors have crept in:
$\mu = \frac12$, $\sigma = 1$, $a=0$, $b = 1$

The blue line is the empirical Monte Carlo pdf, and the red dashed line is the theoretical pdf $h(v)$ above. Looks fine :)

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Besides the reciprocal of the slash distribution (or @Glen_b's "backslash distribution!"), a kind of ratio distribution, I don't know what to call it either, but I'll simulate one version in R.
Since you specify a positive mean of $Y$, I'll use $Y=\mathcal N(7,1)$ so that $\min(Y)>1$ in most samples of $N\le1\rm M$. Of course, other possibilities exist. For instance, any $Y<1$ would expand the range of $\frac X Y$ beyond 1, and any $Y<0$ would of course expand it into negative values. set.seed(1);x=rbeta(10000000,1,1)/rnorm(10000000,7);hist(x,n=length(x)/50000)
(Decrease size for slow computers! Or use runif if you know how!)

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the extreme tails are mucking up the density. The distribution is rather like a Cauchy. (Out of curiosity, why not use runif? It seems more idiomatic and seems also to be faster) –  Glen_b Mar 27 at 5:19
Because I still don't know that much about R, apparently! :) Thanks for the tip! –  Nick Stauner Mar 27 at 6:17
no worries. The difference in speed is not so large, but with 10^7 elements, enough to notice. You may find a histogram worth looking at (hist(x,n=length(x),xlim=c(-10,10))) (about 96% of the distribution seems to be inside those limits) –  Glen_b Mar 27 at 6:22
Wow! Sure enough. Makes these density plots quite misleading I'm afraid! I'll edit in that histogram... –  Nick Stauner Mar 27 at 7:17
Oh, okay. No worries. You may want to make nclass a good deal smaller in that case. I think ideally the bars should be very narrow but not just black lines. –  Glen_b Mar 27 at 9:17