Q.7.45 - A First Course in Probability

Question

If $X_1,X_2,X_3$, and $X_4$ are (pairwise) uncorrelated random variables, each having mean 0 and variance 1 , compute the correlations of

(a) $X_1+X_2$ and $X_2+X_3$

(b) $X_1+X_2$ and $X_3+X_4$.

Step-by-Step Solution

Answer

The correlation of $X_1+X_2$ and $X_2+X_3$ is $\frac{1}{2}$.

The correlation of $X_1+X_2$ and $X_3+X_4$ is $0.$

Step 1: Given Information (Part a)

The Pairwise uncorrelated random variables, each having mean 0 and variance 1 is$=X_1,X_2,X_3,X_4$ 

The correlations of

(a) $X_1+X_2$ and $X_2+X_3=?$

Step 2: Explanation (Part a)

If $X_1,X_2,X_3,X_4$ are (pairwise) uncorrelated

$\Rightarrow Cov\left({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_{\mathrm{j}}\right)=0 \forall i\ne j=1,2,3,4$

$Cov\left(X_1+X_2,X_2+X_3\right)$

$=Cov\left(X_1,X_2\right)+Cov\left(X_1,X_3\right)+Cov\left(X_2,X_2\right)+Cov\left(X_2,X_3\right)$

$=0+0+Var\left({\mathrm{X}}_2\right)+0$

$=1 \left(\because Var\left({\mathrm{X}}_{\mathrm{i}}\right)=1 \forall \mathrm{i}=1,2,3,4\right)$

Step 3: Explanation (Part a)

Calculate the correlation of $X_1+X_2$ and $X_2+X_3,$

$Var\left({\mathrm{X}}_1+{\mathrm{X}}_2\right)=Var\left({\mathrm{X}}_1\right)+Var\left({\mathrm{X}}_2\right)$

$=1+1$

$=2$

$Var\left({\mathrm{X}}_2+{\mathrm{X}}_3\right)=Var\left({\mathrm{X}}_2\right)+Var\left({\mathrm{X}}_3\right)$

$=1+1$

$=2$

$Corr\left(X_1+X_2,X_2+X_3\right)=\frac{Cov\left(X_1+X_2,X_2+X_3\right)}{\sqrt{Var\left(X_1+X_2\right)\sqrt{Var\left(X_2+X_3\right)}}}$

$=\frac{1}{\sqrt{2}\sqrt{2}}$

$=\frac{1}{2}$

Step 4: Final Answer (Part a)

Therefore, the correlation of $X_1+X_2$ and $X_2+X_3$ is $\frac{1}{2}.$

Step 1: Given Information (Part b)

The Pairwise uncorrelated random variables, each having mean 0 and variance 1 is 

$=X_1,X_2,X_3,X_4$

The correlations of

(b) $X_1+X_2$ and $X_3+X_4=?$

Step 2: Explanation (Part b)

$\begin{array}{r}\mathrm{Cov}\left(X_1+X_2,X_3+X_4\right)\end{array}$

$\begin{array}{r}=\mathrm{Cov}\left(X_1,X_3\right)+\mathrm{Cov}\left(X_1,X_4\right)+\mathrm{Cov}\left(X_2,X_3\right)+\mathrm{Cov}\left(X_2,X_4\right)\end{array}$

$=0+0+0+0$

$=0$

$\Rightarrow \mathrm{Corr}\left(X_1+X_2,X_3+X_4\right)=0$

Step 3: Final Answer (Part b)

Hence, the correlation of $X_1+X_2$ and $X_3+X_4$ is $0.$