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 existing answer seems to meet all the requirements in your wish list.  For completeness, I just thought I'd add an extremely simple case matching the main requirement, without using continuity, etc.  This example does not meet your criteria, but it is added to show that it is extremely easy to get distributions that meet  the main requirement (which may be counter-intuitive to some readers).

Example: If $X \sim \text{Bern}(\theta)$ with non-degenerate parameter $0<\theta<1$ you get:
$$\mathbb{P}(X > \mathbb{E}(X)) = \mathbb{P}(X > \theta) = \mathbb{P}(X = 1) = \theta.$$
Taking $\theta = 0.74$ then gives you the required outcome $\mathbb{P}(X > \mathbb{E}(X)) = 0.74$.
