Normal Moments Made Easy

This question focuses on computing low-order moments of a standard normal random variable. The candidate is asked to determine expectations of powers of the variable and to recognize how symmetry of the distribution affects these quantities. It is a clean, self-contained exercise in probability distributions, well suited to testing whether someone can move comfortably between intuitive properties of the normal and concrete numerical expectations, without getting lost in technicalities or excessive calculation.

The solution leans on ideas from moment theory, including recognition of odd versus even powers under a symmetric density and the use of the moment generating function or repeated differentiation to obtain explicit values. An interviewer is looking for fluency in switching between integration, MGFs, and known properties of the normal distribution, as well as the ability to argue clearly about why certain terms vanish. Efficiency of method choice, correctness of algebra, and clarity in explaining the role of symmetry are more important than rote memorization of standard normal moments.

What it tests

The moments of a symmetric probability distribution, such as the standard normal, are governed by the interplay between the function's symmetry and the algebraic structure of the moment (i.e., whether the exponent is odd or even). For any distribution symmetric about zero, all odd moments vanish because the integrand $x^n f(x)$ is an odd function and integrates to zero over the entire real line. Even moments, on the other hand, are nonzero and can be systematically computed using the moment generating function (MGF), which encodes all moments as derivatives at zero. The MGF approach works because differentiation under the integral sign brings down powers of $x$, and evaluating at $t=0$ isolates the desired moment. This principle holds for any distribution where the MGF exists and is especially efficient for the normal distribution due to its closed-form MGF.