问题及解答

Use the result of Exercise 28(Here is Question2505) to show that when $X$ and $Y$ are independent, $\mathrm{Cov}(X,Y)=\mathrm{Corr}(X,Y)=0$.
 

If $X$ and $Y$ are independent, then we have $E(XY)=E(X)\cdot E(Y)$. (See Question2505)

Then, the covariance between $X$ and $Y$ is

\[
\begin{split}
\mathrm{Cov}(X,Y)&=E(XY)-\mu_X\cdot\mu_Y\\
&=E(X)\cdot E(Y)-E(X)\cdot E(Y)\\
&=0.
\end{split}
\]

Therefore, the correlation coefficient of $X$ and $Y$ is

\[
\rho=\rho_{X,Y}=\mathrm{Corr}(X,Y)=\frac{\mathrm{Cov}(X,Y)}{\sigma_X\cdot\sigma_Y}=0.
\]