The correlation between independent random variables is always 0. This is true because the variances of the random variables are well defined.

Detailed Explanation:

1. Definition of Correlation: Correlation measures the linear relationship between two random variables. The correlation coefficient ρ between two random variables X and Y is defined as: ρ = Cov(X,Y) / (σₓ * σᵧ) where Cov(X,Y) is the covariance between X and Y, and σₓ and σᵧ are the standard deviations of X and Y respectively.

2. Independence and Covariance: For independent random variables, the covariance is always 0. This is because: Cov(X,Y) = E[XY] - E[X]E[Y] For independent variables, E[XY] = E[X]E[Y], so Cov(X,Y) = 0.

3. Correlation Calculation: If Cov(X,Y) = 0, then regardless of the standard deviations (which are positive and well-defined as given in the question), the correlation ρ = 0 / (σₓ * σᵧ) = 0.

4. Key Points:

   • Independence implies zero correlation
   • However, zero correlation does NOT necessarily imply independence (it only implies no linear relationship)
   • The condition of "well-defined variances" ensures that the correlation calculation is valid (no division by zero)

Therefore, for independent random variables with well-defined variances, the correlation is always 0.