Question

Let \(X_{1}\) and \(X_{2}\) be independent random variables with nonzero variances. Find the correlation coefficient of \(Y=X_{1} X_{2}\) and \(X_{1}\) in terms of the means and variances of \(X_{1}\) and \(X_{2}\).

Short answer

The correlation coefficient of \(Y=X_{1} X_{2}\) and \(X_{1}\) in terms of the means and variances of \(X_{1}\) and \(X_{2}\) is given as \(\rho_{Y, X_1} = \frac{E(X_2)}{\sqrt{E(X1^2)E(X2^2) - E(X1)^2E(X2)^2}}\).

Step-by-step solution

Compute Covariance

Calculate the covariance between \(Y\) and \(X_1\). By definition, \(Cov(Y, X_1) = E(YX_1) - E(Y)E(X_1)\). Since Y is defined as \(X_1 X_2\), substitute Y in the equation to get \(Cov(X_1 X_2, X_1) = E(X_1^2 X_2) - E(X_1 X_2)E(X_1)\).

Express Covariance in terms of Means and Variances

Go further with the expression obtained: The term \(E(X_1^2 X_2)\) can be expressed as \((E(X_1^2)) (E(X_2))\) since \(X_1\) and \(X_2\) are independent. The term \(E(X_1 X_2)\) is \(E(X_1)E(X_2)\). Substituting these into the equation gives \(Cov(Y, X_1) = (E(X_1^2)) (E(X_2)) - (E(X_1))^2 (E(X_2)) = (E(X_2)) Var(X_1)\), where \(Var(X_1)\) is the variance of \(X_1\).

Compute Correlation Coefficient

The correlation coefficient is defined as \(\rho_{Y, X_1} = \frac{Cov(Y, X_1)}{\sqrt{Var(Y)Var(X_1)}}\). Substituting the equation found in Step 2 into the formula, we obtain \(\rho_{Y, X_1} = \frac{(E(X_2)) Var(X_1)}{\sqrt{Var(Y)Var(X_1)}}\).

Express \(Var(Y)\)

We realise that \(Var(Y)\) can be broken down as follows: \(Var(Y) = E(Y^2)-E(Y)^2\). Since \(Y=X1X2\), we have \(Var(Y) = E((X1X2)^2) - E(X1X2)^2\). This leads to \(Var(Y) = E(X1^2)E(X2^2) - E(X1)^2E(X2)^2\). So as not to complicate matters, we take this expression as the variance of \(Y\).

Conclusion

Insert the previously obtained expressions for our computation back into the correlation coefficient and obtain \(\rho_{Y, X_1} = \frac{(E(X_2)) Var(X_1)}{\sqrt{E(X1^2)E(X2^2) - E(X1)^2E(X2)^2)Var(X_1)}}\). The \(Var(X_1)\) terms in the numerator and the denominator cancel out, leaving \(\rho_{Y, X_1} = \frac{E(X_2)}{\sqrt{E(X1^2)E(X2^2) - E(X1)^2E(X2)^2}}\). This is the final answer.