Uniform - circle

Question

Calculate the probability that $$P(X\le 25,Y\le25)$$ I am unsure how to get started on this question. Any help will be greatly appreciated.

Answer 1

HINT: The probability will be ratio of area of $\{(x,y):x^2+y^2\leq2000^2\}\cap((-\infty,250]\times(-\infty,250])$ to the area of $\{(x,y):x^2+y^2\leq2000^2\}$.

Answer 2

Hint: The equation $$x^2 + y^2 \leq 2000^2$$ describes the area of a circle centered at the origin with radius $2000$.

Since the distribution is uniform, the probability density function is going to be $$f(x, y) = \dfrac{1}{A}$$ over all appropriate values of $x$ and $y$ (which I will leave for you to find), where $A$ is the area of the circle.

From here, all you need to do is integrate $f(x, y)$ appropriately to obtain your desired probability. However, you will need to be cautious about the bounds you choose.