EX1=((( α r1 - r2 ))/(α-1));

(* expectation of X_1^2 *)
EX12 = α ((((r2-r1))^2)/(α-2))-(r2^2)+2 r2 EX1//Simplify

(* the variance of X_1 *)
VarX1=Simplify[ EX12 - (((EX1))^2)]

EX2=((( α r2 - r1 ))/(α-1));

(* the expectation of X_2^2 *)
EX22=α ((((r2-r1))^2)/(α-2))-(r1^2)+2 r1 EX2//Simplify

(* the variance of X_2 *)
VarX2=Simplify[ EX22 - (((EX2))^2)]

(* compute E(X1 X2)  *)
EX1X2 = Simplify[((-1)/2)( α (α+1) ((((r2-r1))^2)/((α-1) (α-2)))-EX12-EX22)]

EX1X2 - EX1 EX2//Simplify

Sqrt[VarX1 VarX2]//Simplify[#,{α>2,r2>r1}]&