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


This is possible if there are very large outliers.
In general, if a distribution has outliers with values that are extremely different from the values of the rest of the distribution, then you are likely to have a mean significantly different from the median value.
So, for instance, a distribution of [-1000, 1, 2, 3] would have a median value of 1.5, a mean value of -248.5, and 75 percent of its distribution would be above the mean value.
