# What are operators that map a number to a probability distribution function called?

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?

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I can't think of a more narrow branch to look in than "statistics". I imagine someone who knows more about the topic will quickly post a better answer than this, but since it is currently unanswered I'll take a stab at the problem.

I had luck googling "assigns a probability density function" in quotes. The idea of a "field" of probability distributions assigned to points of a space sounds pretty natural to me. I also saw "kernel density estimator" appearing several times, so you might look into that.

Lots of the important probability density functions are purely functions of their parameters, for example:

• The normal distribution $\frac{1}{\sigma\sqrt{2\pi}}\exp({-\frac{1}{2}(\frac{x-\mu}{\sigma})^2})$

• the Chi-squared distribution: $\frac{1}{2^{\frac{k}{2}}\Gamma(\frac{k}{2})}x^{(k/2)-1}\exp(-x/2)$

• the t-distribution

It would be interesting to know more about any special properties you want the mapping to have.

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