Explanation

Let's analyze each statement:

Statement I: TRUE

• The sum of two independent normal random variables is indeed normally distributed
• If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²)
• This is a fundamental property of normal distributions

Statement II: FALSE

• The product of two normal random variables is NOT normally distributed
• The product distribution is more complex and follows a different distribution (not normal)
• For example, if X and Y are standard normal, their product follows a distribution with heavier tails

Statement III: FALSE

• The sum of two lognormal random variables is NOT lognormally distributed
• Lognormal distributions are closed under multiplication, not addition
• If X and Y are lognormal, then X + Y does not follow a lognormal distribution

Therefore, only Statement I is correct, making Option A the correct answer.