# Continuous Bivariate Independent Random Variables

Two continuous, independent random variables X and Y each take a random value uniformly distributed between 0 and 8. What is the probability that the difference between them is greater than 4?

Intuitively, I think it's 1/2, because it seems like for every case where the difference is greater than 4, there is a mirrored case where the difference is lesser than 4.

Any ideas on how to prove or am I just wrong?

• There are various different probability distributions on the interval from $0$ to $8$. – Michael Hardy Apr 23 '14 at 18:03
• There are various different dependence relationships beteween probability distributions on the interval from 0 to 8. – Did Apr 23 '14 at 18:04
• You probably want to write are independent. – Did Apr 23 '14 at 18:13
• To solve this, draw an $8\times8$ square: $0 \leq x \leq 8$ and $0 \leq y \leq 8$. Draw lines for $|x-y| = 4$. Measure areas. – user3294068 Apr 23 '14 at 20:46

For this, we know that $x\in\left[0,8\right]$ and $y\in\left[0,8\right]$, so we just need to have $\left|x-y\right|>4$. This only occurs if $x\in\left(4,8\right]$ and $y\in\left[0,4\right)$ or visa versa with $y\in\left(4,8\right]$ and $x\in\left[0,4\right)$.