Finding distribution of distance from origin

Question

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed normally N(0, 1) I want to find the distribution of the distance from the origin to the point of impact. I've tried finding the distributions of $X^2$ and $Y^2$, which I think is $1/(2pix)exp(-x/2)$ I don't then know how to find the distribution of the square root of that. I suspect there's an easier way though. Thanks

Answer 1

What you're looking for is the Rayleigh distribution (distribution of the norm of two centered and independent gaussian RVs) : http://en.wikipedia.org/wiki/Rayleigh_distribution

You might also want to look up the $\chi^2$ distribution (distribution of the squared norm) : http://en.wikipedia.org/wiki/Chi-squared_distribution

Answer 2

Let $R$ denote the distance of the point $(X,Y)$ from the origin. Draw a circle of radius $a$ centered at the origin. Then, $$P\{R \leq a\} = F_R(a) = \iint_{x^2+y^2\leq a}f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy.$$ Convert from rectangular coordinates $(x,y)$ to polar coordinates $(r,\theta)$ hopefully not forgetting the mantra $r\,\mathrm dr\,\mathrm d\theta$ (really, that should be taught and remembered as $r\,\mathrm d\theta\,\mathrm dr$) or changing the limits. Integrate with respect to $\theta$.

Next, put your work aside for a while, and find the derivative (yes, the derivative!) of $-e^{-r^2/2}$. Admire the result for a while. Then turn back to the integral you still need to compute. Don't see how to solve it? Go back to that derivative and admire it some more, and maybe look in your calculus book for The Fundamental Theorem of Calculus.

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Alternatively, after integrating with respect to $\theta$, you might want to write down that, by definition, $$F_R(a) = \int_{-\infty}^a f_R(r)\,\mathrm dr = \int_0^a f_R(r)\,\mathrm dr$$ (the latter integral being valid because $f_R(r)$ must be zero for $r < 0$). Can you deduce anything useful from the two integral formulas that you now have for $F_R(a)$?