Income seems to follow a lognormal distribution within an appropriately defined population (apart for very high earners, who follow a Pareto distribution). Utility (and happiness) are arguably logarithmic in wealth. Ergo, happiness (due to wealth) is normally distributed within a given population! I can’t be the first person to have realized this.

Note that this implies nothing about how income or happiness should be distributed. Also, there are clearly other factors contributing to happiness, but I found this relation interesting.