expected value of $x_{(1)} + x_{(4)} - x_{(2)} - x_{(3)}$

probability

Solution 1:

If a fair $6$-sided die is thrown $4$ times with ordered outcomes $(x)_1\leq(x)_2\leq(x)_3\leq(x)_4$ then $(x)_1$ will have the same distribution as $7-(x)_4$ and $(x)_2$ will have the same distribution as $7-(x)_3$.

From this we conclude that:

$$\mathbb E(x)_1=7-\mathbb E(x)_4\text{ and }\mathbb E(x)_2=7-\mathbb E(x)_3$$

Consequently:$$\mathbb E[(x)_1+(x)_4-(x)_2-(x)_3]=7-7=0$$

Related

Shrinking a triangle keeping all angles the same (for practical application)
Continuous on a set
If $(a+b)(a+c)(b+c)=8abc$ prove $a=b=c$
ask the radius of convergence of power series
Is it possible for a set of well-formed formulas to satisfy a formula and its negation?
Equation of a section plane in hyperbolic paraboloid
How to find an example for a measure which is not continuous from above?
Consequence of concavity
If $N,K$ are two finite groups with isomorphic subgroups, show $ X \leq N \times K $ is normal.
galois group of finite field [duplicate]
Relation between the maximum of a function and mean value of integral
Creating random variables whose pairwise differences are uniformly distributed