Explanation

Let's analyze each statement:

Statement I: "The sum of two independent random normal variables is also an independent random normal variable."

• TRUE: This is a fundamental property of normal distributions. If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²).

Statement II: "The product of two random normal variables is also a random normal variable."

• FALSE: The product of two normal variables is NOT normally distributed. In fact, the product follows a more complex distribution (related to the Bessel function).

Statement III: "The sum of two random lognormal variables is also a random lognormal variable."

• FALSE: The sum of lognormal variables is NOT lognormal. Lognormal distributions are closed under multiplication, not addition.

Therefore, only Statement I is true, which corresponds to option A.