The probability density function $f(x)$ of a random variable $X$ which takes real values is $$f(x) = \frac{1}{3\sqrt{2\pi}} \exp\left(-\frac{x^2}{18}\right), \quad x \in (-\infty, +\infty)$$ Which one of the following statements is correct about the random variable $X$ ?

The general probability density function (PDF) of a normal (Gaussian) random variable $X \sim \mathcal{N}(\mu, \sigma^2)$ is given by:

Comparing this with the given PDF:

We observe:

• $\mu = 0$

• $\sigma = 3$

• $2\sigma^2 = 2(3^2) = 18$

Therefore, $X$ is a normal random variable with mean $\mu = 0$ and variance $\sigma^2 = 9$.

Hence, option (B) is correct.