Probability distribution of maximum of two uniformly distributed random variables

Question

I have two uniformly dirstributed random variables $$r_1 = \mathcal{U}(0,1)$$ and $$r_2 = \mathcal{U}(0,1)$$. I am looking for the probability distribution of $$max\{r_1, r_2\}$$. Is there any known distribution?

Answer 1

Hint

You can't compute it if you don't know the joint ditribution of $(r_1,r_2)$ or if $(r_1,r_2)$ are not independent. If you know the joint distribution or if they are independent, remark that

$$\mathbb P(\max\{r_1,r_2\}\leq x)=\mathbb P(r_1\leq x,r_2\leq x).$$