Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If a point is picked uniformly at random inside a disc of radius $9$, find the CDF and density of the distance $T$ from the center for $0 < t < 9$ .

share|cite|improve this question

We find the cumulative distribution function $F_T(t)$ of $T$. This is the probability that $T\le t$.

Now $T\le t$ if and only if the point picked is inside or on the circle with centre the origin and radius $t$. This circle has area $\pi t^2$. The whole circle has area $\pi(9^2)$. Since our distribution is uniform, we have $$F_T(t)=\Pr(T\le t)=\frac{\pi t^2}{\pi(9^2)}=\frac{t^2}{9^2}$$ when $0\lt t\lt 9$. For completeness, we should add that $F_T(t)=0$ if $t\le 0$, and $F_T(t)=1$ when $t\ge 9$, though this is not asked for.

For the density function $f_T(t)$, differentiate $F_T(t)$ with respect to $t$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.