When the score function is the identity function - A tale of characterizations of the normal distribution

The normal distribution is well-known for several results that it is the only to fulfil. Much less well-known is the fact that many of these characterizations follow from the fact that the derivative of the log-density of the normal distribution is the (negative) identity function. This a priori very simple yet surprising observation allows a deeper understanding of existing characterizations and paves the way for an immediate extension of various seemingly normal-based characterizations to a general density by replacing the (negative) identity function in these results with the derivative of that log-density.