Where can I find examples of a U-Shaped distribution?

A classic example was time-to-failure rates for components such as light bulbs. They typically fail immediately or after a few thousand hours, but rarely in between.

The classical harmonic oscillator is a great example - the probability of being between x + dx is inversely proportional to its velocity there, therefore it's most likely to be near the extrema of the potential. here's a paper that calculates the PDF: http://robinett.phys.psu.edu/qm/INSTRUCTORS/AUTHOR_PAPERS/classical_and_quantum_ajp.pdf the PDF is simply: 1π√x20−x2 1Ï€x02−x2  \frac{1}{\pi\sqrt{ x_{0}^2 - x^2}} \ where x0 is the amplitude of oscillation. here's a plot for an amplitude of 10:

The http://en.wikipedia.org/wiki/L%C3%A9vy_arcsine_law states that the proportion of time that a Brownian motion is positive follows a U-shaped distribution.

Professor David Autor at the MIT Economics Dept. has published studies identifying a recent trend towards low- and high-skilled occupations occurring with movement away from medium-skilled occupations. This results in what he calls a "U-shaped curve" -- c.f. this PDF document from the New America Foundation: http://newamerica.net/sites/newamerica.net/files/policydocs/Losing%20Middle%20America%20-%20Polarization%20-%20Damme%20-%2014%20Sept%2011.pdf On a side note, I didn't realize U-shaped curves were all that rare. Then again, the Central Limit Theorem would have us guess otherwise.

A "real-world" example would be Loss Given Default - an important parameter to measure Credit Risk in financial institutions. http://www.ecb.europa.eu/events/conferences/shared/pdf/net_mar/Session4_Paper1_Memmel_Sachs_Stein.pdf?3b618e15753bcfde10e70679ce53ac37

The http://en.wikipedia.org/wiki/Bathtub_curve which models electronic device failure (like hard drives) is a classic U-shaped distribution, marking the prevalence 0f early life failures and end-of-lifespan failures.  The disk drive industry uses this model heavily. I've never heard of a "U-squared" distribution.  It really makes no sense at all since U is not a variable in this case, but a shape. The only $FOO-squared distributions in common use are Chi-squared and T-squared, and neither bunches at the limits.  Maybe they are trying to coin a new phrase?

Sometimes a U-shaped distribution makes sense for prior beliefs. E.T. Jaynes gives the example of using a Haldine prior to model your (initial) belief about the probability that a solution dissolves in water. If you think the solution very likely to dissolve every time or not at all, then your prior should put most of it's weight at 0 or 1, with little in the middle. BTW the Beta distribution with parameters less than 1 can be a good way to model these distributions.

These distributions are common in dosage relationships for drugs and food.  For example, some but not all studies of red meat consumption v mortality show that a little is better than none but eating a lot is worse.  Alcohol consumption shows similar u-shaped curves in some populations.  Similar relationships obviously hold for drug dosage: the optimum dose is, well, optimum; none or an overdose is worse. Caution: This is not a recommendation to pile up the meat and line up the drinks. The studies that indicate a benefit from red meat and alcohol only find it for the bottom quintile of consumption quantity/frequency.  Other studies have found no benefit reduction in all cause mortality.  High - or even moderate - consumption is definitely riskier.

A common U-Shaped distribution used by life actuaries is the "curve of deaths" i.e. probability of death as a a function of age. Up until the first half of the 20th century, the highest probability of death occurred during the first few months of life; the second highest probability of death occurred during old age. However in the second half of the 20th century, these have swapped and dying of old age is now more common. Still, the probability of dying at either age 1 month or 70 years old is much larger than the probability of dying at 40 years old. http://www.longevitas.co.uk/site/informationmatrix/?tag=curve+of+deaths