Answer: If a random variable X follows a lognormal distribution, ln X is normally distributed.

## 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.

Author: Tanishq Prabhu