Explanation

The correct relationship between normal and lognormal distributions is:

Option A is correct: If a random variable X follows a lognormal distribution, then ln(X) follows a normal distribution. This is the fundamental definition of the lognormal distribution.

Option B is incorrect: If X follows a normal distribution, then ln(X) does NOT follow a lognormal distribution. The lognormal distribution is defined specifically for positive random variables whose logarithm is normally distributed.

Option C is incorrect: The mean and variance of a lognormal distribution are not simply twice those of the corresponding normal distribution. The relationship is more complex:

• If ln(X) ~ N(μ, σ²), then:
  • E[X] = exp(μ + σ²/2)
  • Var(X) = [exp(σ²) - 1] × exp(2μ + σ²)

Option D is incomplete and incorrect: The statement is incomplete and doesn't accurately describe the relationship between the distributions.

Key Points:

• Lognormal distribution: A continuous probability distribution of a random variable whose logarithm is normally distributed
• Transformation: If X ~ Lognormal(μ, σ²), then ln(X) ~ N(μ, σ²)
• Applications: Lognormal distributions are commonly used in finance to model stock prices, as they cannot be negative and exhibit the multiplicative nature of returns.