Finding the probability density function of order statistics

Question

Let $Y_1$ and $Y_2$ be uniformly distributed over the interval $ (0,1) $ then find the probability density function of $ X = \max(Y_1,Y_2) $

Okay in order the find the probability density function of $X$ I need to know that probability density function of the population, right?

How come I can make the assumption that probability density function is uniformly distributed just like the samples drawn from it? I'm asking since I believe that's how we derive the population density function of the population otherwise we wouldn't be able to do the question.

Answer 1

Assuming that $Y_1$ and $Y_2$ are i.i.d, then \begin{align} F_X(x) =& P(X \le x) = P(Y_1 \le x, Y_2 \le x)= P(Y_1 \le x)P(Y_2 \le x)=x^2, \end{align} hence, $$ f_{X}(x)=2x, \quad x\in (0,1). $$

Answer 2

Simply apply the definition order statistics with 2 samples. Then, $X$ is distributed according to

$$f(x) = 2 x$$

with domain of support $x \in (0,1)$