One optimal question

Let M be the alphabet set M={0,1,...,m-1}
Choose x from M set, the probability of choose 0 is p0
the probability of choose 1 is p1
....


Choose y from M set with the same probability

Now trying to calculate
max E(x-y)^2
and min E(x-y)^2


I am doing in this way
Let T = x-y, so the distribution of T is
P_T(t) = \sum_0^{m-1-t} p(t+y)p(y)
f = E(x-y)^2
= \sum_{-(m-1)}^{m-1} t^2 \sum p(t+y)p(y) dy

df/dp(y)

Well I am stuck here.
Thanks a lot!

.

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