(Solution) - Use the following steps to prove that the geometric mean -(2025 Original AI-Free Solution)

Paper Details

Use the following steps to prove that the geometric mean of the product of two random variables X and Y is equal to the product of the geometric means.
17. Write the definition of the geometric mean of the product XY.
18. Use a law of logs to expand the result.
19. Use Theorem 7.4 to break up the expectations.
20. Use a law of exponents and the definition of the geometric mean to prove the result.