Denote $Y_{(1)} = \min(Y_1,Y_2)$ and $Y_{(2)} = \max(Y_1,Y_2)$.
Let $Y_1$ and $Y_2$ be independent and uniformly distributed over the interval $(0, 1)$. Find
$P(2Y_{(1)} < Y_{(2)})$.
Attempted solution:
We know that
$$
f(y_i) =
\begin{cases}
1 & 0<y_i<1\\
0 & else
\end{cases}
$$
Therefore, we determine
$$
F(y_i) =
\begin{cases}
0 & y_i < 0\\
y_i & 0<y_i<1\\
1 & y_i > 1\\
\end{cases}
$$
Using this, we can find the distribution functions for $Y_{(1)}$ and $Y_{(2)}$
$$
Y_{(1)} = 1 – (1-F(y))^2\\
Y_{(2)} = F(y)^2
$$
By differentiating, we get the density functions
$$
f_{Y_{(1)}}(y) = 2(1-y)\\
f_{Y_{(2)}}(y) = 2y
$$
I'm not too sure where to go from here.
Best Answer
Since $a\vee b=(a+b)/2+|a-b|/2$ and $a\wedge b=(a+b)/2-|a-b|/2$,
\begin{align} \mathsf{P}(2Y_{(1)}<Y_{(2)})&=\mathsf{P}(Y_1+Y_2<3|Y_1-Y_2|) \\ &=\mathsf{P}(Y_1+Y_2<3(Y_1-Y_2))+\mathsf{P}(Y_1+Y_2<3(Y_2-Y_1)) \\ &=\mathsf{P}(2Y_2<Y_1)+\mathsf{P}(2Y_1<Y_2)=1/2. \end{align}
$$ \mathsf{P}(2Y_2<Y_1)=\int_0^1\mathsf{P}(Y_2<y/2)dy=\int_0^1(y/2)dy=1/4. $$