I'm a scientist who has stumbled upon an idea that I think might be helpful in my field. I'm looking for information about whether it has been treated in mathematics before – and I would be surprised if it hasn't – and if so, a pointer to some of the results/interesting applications/ideas/etc that have been written about it.
The idea is simple and as follows: higher-order functions that map (ie. take as input) a number to (ie. produce as output) a probability distribution function. The probability distribution function might be parameterised by a function of the number, or it might not be.
In what branches of mathematics are such operators used, and what are they called?