Maximize Covariance from Known Variances

Question

This question asks the candidate to determine the maximum covariance between two random variables given their variances.

Codemia · Accepted Answer

Covariance is a measure of how much two random variables change together. It indicates the direction of the linear relationship between the variables. Variance, on the other hand, measures the dispersion of a single random variable. The maximum covariance can be derived from the variances of the two random variables, which is crucial in understanding their relationship.

Given the variances Var(X) = 3 and Var(Y) = 27, we can use the formula for covariance:

COV(X, Y) = ρ * σ_X * σ_Y

Where ρ is the correlation coefficient, and σ_X and σ_Y are the standard deviations of X and Y respectively. The maximum value of covariance occurs when ρ = 1, indicating a perfect positive correlation. Thus, we can calculate:

- σ_X = √Var(X) = √3
- σ_Y = √Var(Y) = √27

Substituting these values gives:

COV(X, Y) = 1 * √3 * √27 = √81 = 9.

To summarize the calculations:

1. Calculate standard deviations:
   - σ_X = √3
   - σ_Y = √27

2. Use the maximum correlation (ρ = 1):
   - COV(X, Y) = 1 * √3 * √27

3. Result:
   - COV(X, Y) = 9

This shows that the maximum covariance between X and Y, given their variances, is 9.

Understanding the maximum covariance is important in fields like finance and data science, where relationships between variables can inform decision-making. A high covariance indicates that the variables tend to move together, which can be useful for portfolio management, risk assessment, and predictive modeling.

The maximum covariance is achieved under the condition that the two variables are perfectly positively correlated. In real-world scenarios, this condition may not always hold, and understanding the degree of correlation can provide deeper insights into the relationship between variables.

Maximize Covariance from Known Variances

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