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


The probability of being below the mean for a Poisson distribution with $\lambda=1$ is $73.57$6%. So it shouldn't be hard to imagine a continuous distribution that gets that number over $74$% (you asked for above the mean, but that can be solved with a simple transformation). Pareto distributions can easily have more than $74$% of the probability mass on one side of the mean. For instance, $x_m=2, \alpha = 1$ gives $75$% below the mean.
