Select uniformly a point from the annulus $\{(x,y):1\leq x^2+y^2\leq 4\}$. Let R be the distance to $(0,0)$. Give the cdf and pdf of R.
I know that the sample space of choosing a point would be the area of the annulus ($3\pi$) and I know the method of getting the pdf from the cdf or vice versa. I'm just not sure how to start; How would I represent the distance R from the point (x,y) and would this be the cdf or pdf?
Best Answer
By definition of Euclidean Distance, $R=\sqrt{X^2+Y^2}$.
So immediately we know the support is $\{r: 1\leqslant r^2\leqslant 4\}$
Now, since the points are uniformly distributed: the pdf for $R$, at any point $r$ within that support, will be equal to the ratio of the circumference of a circle with radius $r$ to the area of the annulus. Call this $f_R(r)$.
And the CDF will be the integral $\displaystyle F_R(r)=\int_1^r f_R(s)\mathrm d s$ .