Question

Let \(X\) and \(Y\) be identically distributed random variables with variance \(\sigma^2\in(0,\infty)\). Then the correlation coefficient between \(X\) and \(Y\) is

• A. \(1-\dfrac{E(X-Y)^2}{2\sigma^2}\)
• B. \(1-\dfrac{2E(X-Y)^2}{\sigma^2}\)
• C. \(1-\dfrac{E(X-Y)^2}{\sigma^2}\)
• D. \(1-\dfrac{E(X+Y)^2}{\sigma^2}\)

Hint:

Expand E(X-Y)^2 using Var(X-Y) and the fact that identically distributed variables have equal means and variances, then solve for the covariance and divide by sigma squared.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept.
The correlation coefficient between two random variables with the same variance can be written using \(E(X-Y)^2\). We build this connection through the formula for the variance of a difference, together with the fact that identically distributed variables share the same mean and variance.

Step 2: Note that \(X\) and \(Y\) have the same mean and variance.
Since \(X\) and \(Y\) are identically distributed, \[ E(X)=E(Y)=\mu,\qquad \mathrm{Var}(X)=\mathrm{Var}(Y)=\sigma^2 \]

Step 3: Expand \(E(X-Y)^2\).
Since \(E(X-Y)=E(X)-E(Y)=\mu-\mu=0\), we have \[ E(X-Y)^2=\mathrm{Var}(X-Y)+[E(X-Y)]^2=\mathrm{Var}(X-Y) \]

Step 4: Expand \(\mathrm{Var}(X-Y)\).
\[ \mathrm{Var}(X-Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)-2\,\mathrm{Cov}(X,Y)=\sigma^2+\sigma^2-2\,\mathrm{Cov}(X,Y)=2\sigma^2-2\,\mathrm{Cov}(X,Y) \] So \[ E(X-Y)^2=2\sigma^2-2\,\mathrm{Cov}(X,Y) \]

Step 5: Solve for the covariance.
\[ \mathrm{Cov}(X,Y)=\sigma^2-\frac{E(X-Y)^2}{2} \]

Step 6: Write the correlation coefficient.
By definition, \[ \rho(X,Y)=\frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)}\sqrt{\mathrm{Var}(Y)}}=\frac{\mathrm{Cov}(X,Y)}{\sigma\cdot\sigma}=\frac{\mathrm{Cov}(X,Y)}{\sigma^2} \] Substituting the covariance from Step 5: \[ \rho(X,Y)=\frac{\sigma^2-\dfrac{E(X-Y)^2}{2}}{\sigma^2}=1-\frac{E(X-Y)^2}{2\sigma^2} \]

Step 7: Match with the options.
This is exactly option (A). Options (B) and (C) have the wrong constant multiplying \(E(X-Y)^2\) or the wrong power in the denominator, and option (D) uses \(E(X+Y)^2\) instead of \(E(X-Y)^2\), which is not what the algebra gives.

Final Answer:
\[ \boxed{\rho(X,Y)=1-\frac{E(X-Y)^2}{2\sigma^2}} \]