Let $X,Y,Z,V$ be i.i.d continuous random variables in an interval $[a,b]$. What will be the distribution of $\min(X+Y,Y+Z,X+Z,Z+V,X+V,Y+V)$? Assume the distribution of the random variables to be $F(.)$, with $F(a)=0$ and $F(b)=1$.
Let $W=\min(X+Y,Y+Z,X+Z, Z+V,X+V,Y+V)$. We need to find $P(W\leq w)$. This means $P(X+Y \leq w \wedge Y+Z\leq w\wedge X+Z\leq w \wedge Z+V\leq w\wedge X+V\leq w\wedge Y+V\leq w)$. But these sums are not independent, so I wasn't able to proceed further.
What should be the general approach for such problems?