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For a random variable X and a constant a, show that $Var(aX) = a^2V ar(X)$

    Knowing that $$ Var(X) = E[(X-E(X))^2] = E(X^2)-[E(X)]^2$$

    Then:

    $$
    Var(aX) = E[(aX)^2] - [E(aX)]^2
    $$

    $$
    Var(aX) = E[a^2X^2] - [aE(X)]^2
    $$

    $$
    Var(aX) = a^2E[X^2] - a^2[E(X)]^2
    $$

    $$
    Var(aX) = a^2[E(X^2) - E(X)^2]
    $$

    $$
    Var(aX) = a^2Var(X)
    $$

2.  For random variables X and Y,show that $Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)$.

    Knowing also that

    $$
    Cov(X, Y) = E[(X-E(X))(Y-E(Y))] = E(X,Y) - E(X)E(Y)
    $$

    Then:

    $$
    Var(X + Y) = E[(X + Y-E(X+Y))^2]
    $$

    $$
    Var(X+Y) = E[(X +Y - E(X) -E(Y))^2]
    $$

    $$
    Var(X + Y) = E[[(X-E(X)) + (Y-E(Y))]^2]
    $$

    $$
    Var(X + Y) = E[(X - E(X))^2 + (Y-E(Y))^2+2(X-E(X))(Y-E(Y))]
    $$

    $$
    Var(X + Y) = E[(X - E(X))^2] + E[(Y-E(Y))^2]+2E[(X-E(X))(Y-E(Y))]
    $$

    $$
    Var(X + Y) = Var(X) + Var(Y) +2Cov(X,Y)
    $$

3.  For random variables X and Y , show that $Cov(X, Y ) = E(XY ) − E(X)E(Y )$.

    $$
    Cov(X,Y)=E[(X-E(X))(Y-E(Y))]
    $$

    $$
    Cov(X,Y)=E[XY-XE(Y)-YE(X)+E(X)E(Y)]
    $$

    $$
    Cov(X,Y)=E[XY]-E[XE(Y)]-E[YE(X)]+E(X)E(Y)
    $$

    $$
    Cov(X,Y)=E[XY]-E(X)E(Y)-E(Y)E(X)+E(X)E(Y)
    $$

    $$
    Cov(X, Y ) = E(XY ) − E(X)E(Y)
    $$

4.  For random variables X and Y and constants a and b, show that $Cov(ax, bY ) = abCov(X, Y )$.

    $$
    Cov(aX, bY ) = E[(aX)(bY)]-E(aX)E(bY)
    $$

    $$
    Cov(aX, bY ) = E[(abXY)]-E(aX)E(bY)
    $$

    $$
    Cov(aX, bY ) = abE[XY]-[aE(X)[bE(Y)]
    $$

    $$
    Cov(aX, bY ) = ab[E(XY)-E(X)E(Y)]
    $$

    $$
    Cov(aX,bY) = abCov(X,Y)
    $$