0

Let the point $(u, v)$ be chosen randomly in the $[0, 1] \times [0, 1]$ square. Let $X$ be the random variable that associates to point $(u, v)$ to the number $(u + v)$. Obtain the distribution function of $X$.

Thoughts:

$0 \leq u \leq 1$, and $0 \leq v \leq 1$. Therefore $0 \leq u + v \leq 2$. But $X = u + v$, so we have $0 \leq X \leq 2$. I'm not sure how to take it from here, but I think that

$$ F_X = \frac{x}{2}, ~~ 0 \leq x \leq 2 $$

Noa Even
  • 2,801
Sigma
  • 1,836
  • 15
  • 35
  • The uniform distribution on $[0,1]^2$ is the product of two independent uniform distributions on $[0, 1]$. In particular, $X = U+V$ where $U, V$ are i.i.d Uniform($[0,1]$). Do you know how to find the CDF of the sum of two independent random variables? (Another hint: $X$ is itself not uniformly distributed, which is your current guess.) – snar Jun 19 '19 at 16:38
  • @snar Thanks for the insight. And yes I do know convolutions. – Sigma Jun 19 '19 at 16:46

0 Answers0