The probability density function \(f(x)\) of a random variable \(X\) which takes real values is
\(f(x) = \dfrac{1}{3\sqrt{2\pi}}\exp\left(-\dfrac{x^{2}}{18}\right)\), \(x \in (-\infty, +\infty)\)
Which one of the following statements is correct about the random variable \(X\) ?
GATE CS - 2026,
Answer: B
X is a normal random variable.
The normal density with mean μ and standard deviation σ is
\[ \begin{aligned} f(x) &= \frac{1}{\sigma\sqrt{2\pi}} \\ &\quad \times \exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right) \end{aligned} \]
With μ = 0 and σ = 3, the factor in front is 1/(3√(2π)) and the exponent is -x2/18, exactly the given function, defined on the whole real line.
An exponential density is zero for negative x and decays as e-λx, not e-x2. A Poisson variable takes only non-negative integer values and has a probability mass function, not a density. A uniform density is constant on a bounded interval.
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