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 beta distribution should satisfy your criteria. With first shape parameter of 1.0 and second shape parameter of 0.14, the average is 0.8772 while 74.56 % of the probability is concentrated above that.
Here is a graph of the pdf: 
