# Squiggle invariants

Here are some property tests for squiggle. I am testing mostly for the mean and the standard deviation. I know that squiggle doesn't yet have functions for the standard deviation, but they could be added.

The keywords to search for are "[algebra of random variables](https://wikiless.org/wiki/Algebra_of_random_variables?lang=en)". 

## Means and standard deviations
### Sums

$$ mean(f+g) = mean(f) + mean(g) $$

$$ std(f+g) = \sqrt{std(f)^2 + std(g)^2} $$

In the case of normal distributions,

$$ mean(normal(a,b) + normal(c,d)) = mean(normal(a+c, \sqrt{b^2 + d^2})) $$

### Subtractions

$$ mean(f-g) = mean(f) - mean(g) $$

$$ std(f-g) = \sqrt{std(f)^2 + std(g)^2} $$

### Multiplications

$$ mean(f \cdot g) =  mean(f) \cdot mean(g) $$

$$ std(f \cdot g) = \sqrt{ (std(f)^2 + mean(f)) \cdot (std(g)^2 + mean(g)) - (mean(f) \cdot mean(g))^2} $$

### Divisions

Divisions are tricky, and in general we don't have good expressions to characterize properties of ratios. In particular, the ratio of two normals is a Cauchy distribution, which doesn't have to have a mean.

# To do:

- Provide sources or derivations, useful as this document becomes more complicated
- Provide definitions for the probability density function, exponential, inverse, log, etc.
- Provide at least some tests for division
- See if playing around with characteristic functions turns out anything useful

## Probability density functions

TODO

## Cumulative density functions 

TODO

## Inverse cumulative density functions

TODO