问题及解答

• Show that $\mathrm{Cov}(X,Y+Z)=\mathrm{Cov}(X,Y)+\mathrm{Cov}(X,Z)$.
• Let $X_1$ and $X_2$ be quantitative and verbal scores on one aptitude exam and let $Y_1$ and $Y_2$ be corresponding scores on another exam. If $\mathrm{Cov}(X_1,Y_1)=5$, $\mathrm{Cov}(X_1,Y_2)=1$, $\mathrm{Cov}(X_2,Y_1)=2$, and $\mathrm{Cov}(X_2,Y_2)=8$, what is the covariance between the two total scores $X_1+X_2$ and $Y_1+Y_2$?
   

(1)

By the property of covariance, $\mathrm{Cov}(X,Y)=E(XY)-\mu_X\cdot\mu_Y$, we have

\[
\begin{split}
\mathrm{Cov}(X,Y+Z)&=E[X(Y+Z)]-\mu_X\cdot\mu_{Y+Z}\\
&=E[XY+XZ]-\mu_X\cdot(\mu_Y+\mu_Z)\\
&=E(XY)+E(XZ)-\mu_X\cdot\mu_Y-\mu_X\cdot\mu_Z\\
&=\Bigl(E(XY)-\mu_X\cdot\mu_Y\Bigr)+\Bigl(E(XZ)-\mu_X\cdot\mu_Z\Bigr)\\
&=\mathrm{Cov}(X,Y)+\mathrm{Cov}(X,Z).
\end{split}
\]

And by definition or the property above, we know that the covariance is symmetry: $\mathrm{Cov}(X,Y)=\mathrm{Cor}(Y,X)$.

Hence, we have

\[
\begin{split}
\mathrm{Cov}(X+Y,Z)&=\mathrm{Cov}(Z,X+Y)\\
&=\mathrm{Cov}(Z,X)+\mathrm{Cov}(Z,Y)\\
&=\mathrm{Cov}(X,Z)+\mathrm{Cov}(Y,Z).
\end{split}
\]

(2)

\[
\begin{split}
\mathrm{Cov}(X_1+X_2,Y_1+Y_2)&=\mathrm{Cov}(X_1+X_2,Y_1)+\mathrm{Cov}(X_1+X_2,Y_2)\\
&=\mathrm{Cov}(X_1,Y_1)+\mathrm{Cov}(X_2,Y_1)+\mathrm{Cov}(X_1,Y_2)+\mathrm{Cov}(X_2,Y_2)\\
&=5+1+2+8\\
&=16
\end{split}
\]