Suppose that $Y\_1, Y\_2$ are two independent observations from the following distribution:
$$
f\_\theta(y) = \begin{cases} \dfrac{3y^2}{\theta^3} &\text{for } 0 <y \leq \theta, \ \theta>0 \\[8pt]
0 & \text{otherwise}\end{cases} 
$$

I am trying to find $E(Y\_1 \mid \max(Y\_1, Y\_2))$, and am not sure how to do this. I can directly compute the integral but I still need to find the joint distribution of $Y\_1$ and $\max(Y\_1,Y\_2)$, which I am not sure how to derive.