Question

Question: Show that \(V(X + Y) = V(X) + V(Y) + 2{\mathop{\rm Cov}\nolimits} (X,Y).\)

Short answer

Answer

The resultant answer \(V(X + Y) = V(X) + V(Y) + 2{\mathop{\rm Cov}\nolimits} (X,Y)\)is proved

Step-by-step solution

 Step 1: Given data

The given data is \(V(X + Y) = V(X) + V(Y) + 2{\mathop{\rm Cov}\nolimits} (X,Y)\).

Concept of Covariance

Covariance is:\({\mathop{\rm Cov}\nolimits} (X,Y) = E((X - E(X))(Y - E(Y)))\).

Proof the expression

Expression is:

\(\begin{aligned}{}{\rm{V}}({\rm{X}} + Y) &= E\left( {{{(X + Y)}^2}} \right) - {(E(X + Y))^2}\\ &= E\left( {{X^2} + {Y^2} + 2XY} \right) - {(E(X) + E(Y))^2}\\ &= E\left( {{X^2}} \right) + E\left( {{Y^2}} \right) + 2E(XY) - {(E(X))^2} - {(E(Y))^2} - 2E(X)E(Y)\\ &= V(X) + V(Y) + 2{\mathop{\rm Cov}\nolimits} (X,Y)\end{aligned}\)