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Var[a(X^2)+b] = E[ ( a(X^2) + b - (aE[X^2] + b ) )^2 ]
= E[ ( a(X^2) + b -aE[X^2] - b )^2 ]
= E[ ( a(X^2) - aE[X^2] )^2]
= E[ (a^2) ( (X^2) -E[X^2] )^2 ]
= E[ ( (X^2) - E[X^2] ) ( (X^2) - E[X^2] ) ]
= (a^2) E[ (X^4) -2(X^2)E[X^2] + ( E[X^2] )^2]
= (a^2) { E[X^4] -2E[X^2]E[X^2] + ( E[X^2] )^2 }
= (a^2) { E[X^4] -2( E[X^2] )^2 + ( E[X^2] )^2 }
= (a^2) { E[X^4] - ( E[X^2] )^2 }
= (a^2) Var[X^2]
*however, we still do not know what Var[X^2] is....
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