1. ## Probability

Let x and y be uniformly distributed independent random variables over [0,1]. What is the probability the distance between x and y is less than 1/2?

2. This can be a geometry problem, since the distributions are uniform.

You need $p(|x-y| < \frac{1}{2})$

For x > y, we have $x - y < \frac{1}{2}$ or $y > x - \frac{1}{2}$

This cuts off a corner.

You do x < y to cut off the other corner. You're almost done.

3. Originally Posted by mathman88
Let x and y be uniformly distributed independent random variables over [0,1]. What is the probability the distance between x and y is less than 1/2?
Unit square:

Area of shaded region = 3/4.