ansTheorem 9.3. Let X and Y be independent random variables with finitevariances, and a, b ER. ThenVar(aX) = a²Var (X),Var (X+Y) = VarX + Var Y,Var (ax + bY) = a²VarX + b²Var Y.sercise Prove the theorem.Remark 9.2. Independence is sufficient for the variance of the sum to be equal tothe sum of the variances, but not necessary.Remark 9.3. Linearity should not hold, since variance is a quadratic quantity.Remark 9.4. Note, in particular, that Var (-X) = Var(X). This is as expected,since switching the sign should not alter the spread of the distribution.

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ans Theorem 9.3. Let X and Y be independent random variables with finite variances, and a, b ER. Then Var(aX) = a²Var (X), Var (X + Y) = VarX + Var Y, Var (ax + bY) = a²VarX+ b²Var Y.