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

I have a word problem regarding the cdf and I am struggling to get my head round the idea. It goes like this:

A dart is thrown at a circular target with radius $\alpha$. The dart always hits the target and The probability that the dart hits any particular region of the target is proportional to the area of that region. Let $R$ be the distance between the target centre and the point the dart hits. I must find the cdf for $R$.

I know that the point of contact is uniformly distributed over the target but that is basically it. If someone can run through how I would do this, it would help greatly.

share|cite|improve this question
$cdf=F(r)=P(R\leq r)$ – hhsaffar Nov 28 '13 at 9:39
Can you find $P(R \leq r)$? – hhsaffar Nov 28 '13 at 9:41
Can you expand on that please? How would I consider that for $r<0$, $0 \le r \le \alpha$ and $r> \alpha$? – Stuart Nov 28 '13 at 9:43
if $r>\alpha$ then $R\leq r$ always hold so $P(R \leq r)=?$ – hhsaffar Nov 28 '13 at 9:44
Got you. So for the case $0 \le r \le \alpha$ what would I do and how would I join these facts up to find $P(R \le r)$? – Stuart Nov 28 '13 at 9:51

$ F(r) = P(R \leq r)= \begin{cases} 0 & r<0\\ \frac{r^2}{\alpha^2} & 0 \leq r \leq \alpha \\ 1 & r>\alpha \end{cases}$

For $0 \leq r \leq \alpha$ the probability is $\frac{r^2\pi}{\alpha^2\pi}=\frac{r^2}{\alpha^2}$

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.