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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.

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$cdf=F(r)=P(R\leq r)$ –  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
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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

1 Answer 1

$ 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 $\frac{r^2\pi}{\alpha^2\pi}=\frac{r^2}{\alpha^2}$

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