Cumulative distribution function picture problem

Question

this is from an past exam paper I got for part 1: O≤X≤r = x^2*π /r^2*π and how do I do part 2 is that right, also I don't really understand the question, when it talks about circle in the question, does it actually means within R or r

Answer 1

(i): We're told that $P(X \leq r) = \lambda r^2$, for some constant $\lambda$. Assuming that whoever throws the darts is skilled enough to never miss the disc entirely, we must have $P(X \leq R) = 1 = \lambda R^2$ and it follows that $\lambda = \frac{1}{R^2}$. The CDF of $X$ is thus $$ F_X(r) = P(X \leq r) = \frac{r^2}{R^2} \text{.} $$

(ii): The score $S$ is $10 - i$ if $X$ lies within $\left(R\frac{i}{10}, R\frac{i+1}{10}\right]$ (we're not told what happens if we hit the bounds exactly, so whether we pick open or closed bounds is arbitrary. It won't change the result because hitting the bounds exactly has probability 0). It follows that $$ P(S = 9) = P\left(\frac{R}{10} < X \leq \frac{2R}{10}\right) = F_X(\tfrac{R}{5}) - F_X(\tfrac{R}{10}) = \frac{\frac{R^2}{25} - \frac{R^2}{100}}{R^2} = \frac{3}{100} \text{.} $$