Expectation of the distance to the center and to the boundary of a point in a circle

Question

We randomly choose a point inside a circle of radius 1. Let $X$ be the distance of the point to the center of the circle and $Y$ be the distance of the point to the circle boundary. What is the expected value of $min(X,Y)$?

What I know is that $Y$ could be rewritten as $1-X$, yet could someone show the derivation of the answer? Thanks!

Answer 1

Guide:

• Find the CDF $F_X$ of $X$.
• Find the PDF $f_X$ of $X$ by differentiating the CDF $F_X$.
• Calculate $\mathbb E\min(X,1-X)=\int_0^1\min (x,1-x)f_X(x)dx$.