For historical reasons, I am forced to take my examples from
high energy physics.
Take M happy smiling counters---all arranged in a row:
Suppose they count random events that fall in their province.
Then each of them has a parameter associated with
them such that distribution of number of events in counter is a
Poisson distribution with parameter (t is time).
associated with
them such that distribution of number 
of events in counter is a
Poisson distribution with parameter 
(t is time).