Expected Absolute Difference for Normals

This quant interview question is about understanding how expectations behave for transformed Gaussian variables, a core topic in probability and expected value. It takes a simple-looking normal setup and adds an absolute value and a specific functional form for the answer, forcing you to combine distributional insight with algebraic care. In a quant prep context, it checks whether you really know your normal distribution properties beyond rote formulas.

It trains comfort with independent normals, distributions of linear combinations, and how expectations interact with symmetry and scaling. More conceptually, it sharpens your sense of how absolute values change moments, and how to manipulate expressions into a clean, parameterized form. This is exactly the sort of probabilistic maturity that strong quant interviews probe.

In quant interviews, this matters because it tests fast, rigorous reasoning about continuous distributions, not just plug-and-chug. Successful candidates in trading and research roles must move fluently between distribution properties, expectations, and transformations. A question like this reveals whether your quant prep has gone deep enough to handle similar twists under time pressure in real interviews.

What it tests

When dealing with the absolute value of the difference between two independent normal random variables, the key is to recognize that their difference is itself normally distributed, with mean equal to the difference of the means and variance equal to the sum of the variances. This is a consequence of the properties of independent random variables: variances add, and the sum (or difference) of normals is normal. The expectation of the absolute value of a normal variable (its mean absolute deviation from the mean) is always proportional to its standard deviation, specifically $\sqrt{2/\pi}$ times the standard deviation. This proportionality arises from the symmetry and scaling properties of the normal distribution, and the computation reduces to a standard integral that is independent of the mean (for mean zero) and scales linearly with the standard deviation.