There are many examples to this. Some of the examples are as
followed:
The number of accidents per hour happening at a traffic
crossing.
Waiting time for a customer in a queue. This is expressed as an
exponential distribution
Complex financial instruments valuation from future cashflows:
Here non normal distributions are used as the distribution cannot
have negative values ( lower bounded ) and also, the tail values
are expected to be more likely (fatter tails ) than in normal
distribution.