# Probability functions with 2 random variables

I really dont understand this problem, I was hoping someone could help me out with this and explain how to do this. It goes like this: Suppose you throw a dart at a circular target with a 10 cm radius. Assume that you always hit the target and that the coordinates you hit on the dartboard are chosen at random. Locate the origin at the center of the circle and denote points on the target area by their polar coordinates $(r,\theta)$. Thus $r$ is the distance of the point to the center of the circle and $\theta$ is the angle with the $x$-direction, measures in a counterclockwise direction, and expressed in degrees.

i) Find an expression for the distribution function $F_{R,\theta}(r,\theta)$. Then find the expression for the distribution functions $F_R(r)$ and $F_\theta(\theta)$.

ii) Are the random variables $R$ and $\theta$ independent?

I know its a lot and its long but I am really stuck, so any help would be appreciated greatly.

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## 1 Answer

What is a distribution function? As I remember, by definition, it is $$F_{R,\Theta}(r,\vartheta):=P(R<r,\Theta<\vartheta)$$ that is, assumed that we start to measure the angle from the positive x-axis, say, this is the (area of a sector of angle $0..\vartheta$ and radius $r$)/(area of whole disk). Whereas, $$F_R(r)=P(R<r),\quad F_\Theta(\vartheta)=P(\Theta<\vartheta)$$ are ratios of full disk and full sector. Being independent means that $$F_R(r)F_\Theta(\vartheta) = F_{R,\Theta}(r,\vartheta)\text{ for all }r,\vartheta.$$ I bet they are.

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