Example distribution where 74% of probability is above the mean Watching Why You Should Want Driverless Cars On Roads Now, at 8:14 Derek Muller claims:

Surveys show 74 % of people believe they are above average drivers.

This claim motivates my question, but some clarification is needed. I am not asking for someone to source out the statistic nor to address the empirical claim. Rather, I would like a concrete and simple example of my intuition that it is mathematically possible. I've put together a wish list of additional criteria:

*

*Has a 'clean' algebraic expression for its probability density function or probability mass function (i.e. not just a constructed dataset or a 'tortured piecewise function')

*Support is single-variable

*Support is continuous or contiguous

*If support is discrete, each bin with nonzero probability mass must have one or two adjacent bins with nonzero probability mass

*Distribution is unimodal


A simple real world example would be the number of legs that people have. The huge majority have two legs, nobody has three legs, some few people have one leg or none. The average is just below 2, and the huge majority is a tiny bit above average.
The opposite: Number of citizenships that people have. The huge majority have one, a very small number have none, and a bit more have two or more. The average is just a bit over 1, and the huge majority is below average.
Profit from a risk-only life insurance for the insurance company: If you pay for a life insurance that pays on death only, there will be no payout for most people, with a small profit for the insurance company. A small number create a huge loss. So the majority will create much higher than average profit. This is the situation from another answer, with huge negative outliers, only with a real example.
