Suppose there is a target shooting game on circle of radius $3$. Think of the result of the shooting as a random experiment, for simplicity, we suppose the hit will always impact on the circle of radius $3$. We put the center of the circle at the origin of $\mathbb R^2$, the sample space of the experiment will be $\Omega=\{(x,y): x^2+y^2<9\}$. Let $\mathcal F$ be the $\sigma$-algebra of the borel sets of $\mathbb R^2$. Assume that the probability of a dart being hit in a certain region $A$ is proportional to the its area $|A|$. That means $$P(A)=\dfrac{|A \cap \Omega|}{9\pi}$$ Suppose the score obtained is $3$ minus the distance from the hit to the center. Call $Y$ to the random variable $$Y=3-\sqrt{x^2+y^2}$$
Find the cumulative distribution function of the random variable $Y$.
I am having some doubts with the problem, so I'll write what I've did. $$F_Y(z)=P(Y\leq z)=P(3-\sqrt{x^2+y^2}\leq y)=P(3-z\leq \sqrt{x^2+y^2})$$ If $(x,y) \in \Omega$, the probability I am looking for is the probability of $(x,y)$ being in the border or outside the disk of radius $3-z$. If I denote that region by $A$, then $P(A)=1-P(A^c)=1-P(D_{3-z})$, where $D_{3-z}$ is the circle of radius $3-z$, so $$P(A)=1-\dfrac{|D_{3-z}\cap \Omega|}{9\pi}.$$
So $F_Y(z)=1-\dfrac{|D_{3-z}\cap \Omega|}{9\pi}$. Is this correct? I would appreciate if someone could take a look at the problem and correct the answer.
Your calculations are correct. In practice, since you decided to express the cdf according to the distance $z$ between the hit and the circumference with radius $3$, and taking into account that $z$ also corresponds to the score, the function simply states that the cumulative probability of getting a score $\leq z$ is zero for $z=0$, increases in the interval $0<z<3$ according to a quadratic relation (after some simplifications we can get $\frac{2}{3}z-\frac{z^2}{9}$) and is $1$ for $z=3$. Also note that the corresponding pdf in this interval decreases according to a negative linear relation, reflecting the probability of hitting a point on a circumference with radius $3-z$.