Question

Let \(X\), \(N\), \(Y\) and \(Z\) be random variables. The variables \(X\) and \(N\) are independent of each other. \(X\) is uniformly distributed between \(-1\) and \(1\); \(N\) follows Normal distribution with zero mean and unity variance.
\(Y\) and \(Z\) are defined as \(Y=X+N\) and \(Z=X^2+N\).
Which of the following pairs represents the values of correlation between \(X\) and \(Y\), and that between \(X\) and \(Z\)?

• A. \(\dfrac{1}{3}\) and \(0\)
• B. \(\dfrac{1}{3}\) and \(\dfrac{1}{9}\)
• C. \(\dfrac{1}{3}\) and \(\dfrac{1}{3}\)
• D. \(1\) and \(0\)

Hint:

Use Corr(U,V) = E[UV] - E[U]E[V]; the odd moment E[X^3] of a symmetric uniform variable is zero.

Answer 1

The Correct Option is A

Step 1: Recall the definition of correlation (covariance) between two random variables.
For two random variables \(U\) and \(V\), the correlation used here is the covariance
\[ \text{Corr}(U,V)=E[UV]-E[U]E[V] \]

Step 2: Note the basic facts about X and N.
\(X\) is uniform on \([-1,1]\), so
\[ E[X]=0,\qquad \text{Var}(X)=\frac{(1-(-1))^2}{12}=\frac{4}{12}=\frac{1}{3} \]
Because the density of \(X\) is symmetric about \(0\), every odd moment of \(X\) is zero, so
\[ E[X^3]=0 \]
\(N\) has zero mean, and \(X\), \(N\) are independent, so
\[ E[XN]=E[X]E[N]=0,\qquad E[X^2N]=E[X^2]E[N]=0 \]

Step 3: Find the correlation between X and Y.
\[ \text{Corr}(X,Y)=E[XY]-E[X]E[Y]=E[X(X+N)]-0=E[X^2]+E[XN] \]
Since \(E[X]=0\), \(E[X^2]=\text{Var}(X)=\frac{1}{3}\), and \(E[XN]=0\):
\[ \text{Corr}(X,Y)=\frac{1}{3}+0=\frac{1}{3} \]

Step 4: Find the correlation between X and Z.
\[ \text{Corr}(X,Z)=E[XZ]-E[X]E[Z]=E[X(X^2+N)]-0=E[X^3]+E[XN] \]
Since \(E[X^3]=0\) (odd moment of a symmetric distribution) and \(E[XN]=0\) (independence):
\[ \text{Corr}(X,Z)=0+0=0 \]

Step 5: Analyze the options.

(A) 1/3 and 0: Matches both computed values exactly. Correct.

(B) 1/3 and 1/9: The first value is right but the second should be \(0\), not \(\frac{1}{9}\), since the odd moment \(E[X^3]\) vanishes. Incorrect.

(C) 1/3 and 1/3: Would only hold if \(E[X^3]\) equalled \(\frac{1}{3}\), but it is zero by symmetry. Incorrect.

(D) 1 and 0: The correlation between X and Y is \(\text{Var}(X)=\frac{1}{3}\), not \(1\), since \(X\) does not have unit variance. Incorrect.

Step 6: Final conclusion.
Therefore, the required pair of correlation values is \[ \boxed{\frac{1}{3}\text{ and }0} \]