Suppose G is a lottery where the payoff is equal to -1 with 0.5 probability and 1 with 0.5 probability. I'm trying to show that G is a mean preserving spread of a lottery with uniform distribution on [-1,1] and (stronger) that given any c.d.f. on [-1,1], G is a mean preserving spread of that c.d.f. Any help on either problem would be greatly appreciated.
I understand intuitively how this is true, but I'm having a hard time formalizing this into a proof. I haven't seen an example of how to show a lottery is a mean preserving spread of another.