Conditional Expectation for Normal X|X

This question focuses on computing a conditional expectation for a continuous random variable with a normal distribution, under the condition that the variable lies in a particular region of its support. The setup is the standard normal distribution, restricted to values on one side of the real line, and asks for the expected value under this restriction. It is a prototypical example of working with truncated normal distributions and conditional expectations given events of positive probability on the real line.

To solve it, a candidate must translate the conditional expectation definition into an integral with a renormalized density, recognize that the denominator is the probability of the conditioning event, and correctly handle the standard normal density and symmetry. The question leans on understanding density functions, properties of the normal distribution, and basic measure-theoretic intuition about conditioning on events. Interviewers watch for comfort with setting up integrals for expectations, using symmetry arguments appropriately, and avoiding common mistakes like forgetting to renormalize or misinterpreting the conditioning event.

What it tests

Whenever you compute a conditional expectation over a subset of the support of a continuous random variable, you are essentially reweighting the original distribution so that it becomes a new probability measure restricted to that subset. The conditional expectation is then the mean of this truncated distribution. For symmetric distributions like the standard normal, conditioning on a half-line (such as $X > 0$) leverages the symmetry and the properties of the density function, often simplifying the denominator (the probability of the event) and sometimes the numerator (the expected value over the event). The key is that the conditional expectation is not just the mean over the region, but the mean with respect to the renormalized density. This principle holds because probability measures must sum (or integrate) to 1, so restricting to a subset requires dividing by the probability of that subset, ensuring the new density is valid.