Answer-first summary for fast verification
Answer: I only
## 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.
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Which of the following statements are TRUE? I. The sum of two independent random normal variables is also an independent random normal variable. II. The product of two random normal variables is also a random normal variable. III. The sum of two random lognormal variables is also a random lognormal variable.
A
I only
B
II only
C
III only
D
I and II
E
II and III
F
None of the above