A quantity $X$ is compute by the following method:
a.) Generate two independent uniform random variables $u$, $v$.
b.) If $u^2 + v^2 \geq 1$, return to step $1$. Otherwise set $X = u$.
Find the c.d.f $F(x)$ for $X$. How many times will step $1$ be performed.
Hint: Draw a quarter of a unit circle, inside a unit circle. Pick an arbitrary $x$ on the $x$-axis. Find the probability that $\mathbb{P}\{X\leq x\}$ using geometric arguments via areas.
THoughts: drawing a quarter of a unit circle inside a unit circle then picking an arbitrary $x$ then $$P\{X\leq x\} = \frac{x}{\pi}$$ But I don't think that is right I am a bit lost on this problem. Any suggestions are greatly appreciated.
You are rejection sampling from the uniform distribution on the unit square to get the uniform distribution on the quarter unit circle, and then taking $X$ coordinate. So you need the probability that $X<x$ for $X$ a random sample on the quarter unit circle. This is the fraction of area to the left of $x$ in the quarter unit circle, which is given by $$ P(X\le x) = \frac{1}{\pi/4}\int_{0}^x\sqrt{1-x'^2}dx'.$$
As for how many times the step is going to be performed, that depends on how many samples you take. Your acceptance probability is $\pi/4 $ so on average it will take $4/\pi$ repetitions of step one per sample.