for a measurable function
on a probability space
(i.e. a measure space
of total measure 1
Random variables are the formalisation of the concept of the result of an experiment. You have a system with a random element (your probability space); you perform a measurement. The result is a deterministic function of the state of your system, but your system itself is random. So the result is a random variable.
- The number of heads in a sequence of 163 coin tosses.
- The average value of the results from rolling 17 dice (you'd expect this to be roughly 3.5, but since you're dividing by an odd number you'll never get this number!).