Let $x$ be drawn from the uniform distribution on $[0,1]$.
$x$ parameterises the Triangular distribution $Y$ with support $[0,1]$.
I.e., $$ f_Y(y_i | X = x) = \begin{cases} \frac{2y_i}{x} \quad \mbox{if} \quad y_i < x \\ 2 \quad \mbox{if} \quad y_i = x \\ \frac{2(1-y_i)}{1-x} \quad \mbox{if} \quad y_i > x \end{cases} $$
From $Y$, we draw $n$ random variables $y_i$, with $i = 1, 2, \dots, n$.
I would like to find the unconditional joint density of $y_1$ and the event that $y_1$ is the largest of the $n$ draws.
I have tried the following:
\begin{align*} f_{Y, Y^{max}} = \int_0^1 f_{Y | X}(y_1 | X = x) \Pr[y_1 > \max\{y_2, y_3, \dots, y_n\}] d x \end{align*}
Note that $f_X(x) =1$ everywhere on $[0,1]$
But I am confused whether this is admissible. I have also tried the simpler thing of doing this with $n = 3$ but obtain odd results.