## Density Rain

Illustrates the concept of a probability density function using a Beta distribution.

Drag your mouse to create the desired Beta distribution, then press T to “throw darts” and then D to “rain” the darts down onto the horizontal axis.

Scroll down below the applet for more background and details.

## Details

The process of throwing darts (T) involves using data augmentation to sample value of $X$, the Beta random variable of interest. Let $Y$ be a uniform random variable conditional on the value of $X$ taking on any value between 0 and the height of the density function. Throwing a dart uniformly within the area underneath the density function is equivalent to jointly sampling a point $(X,Y)$ from the joint probability distribution $p(X,Y)$.

The process of raining darts (D) involves integrating over all possible values of $Y$ for every value of $X$, yielding a valid sample from the distribution of $X$.

$p(x) = \int p(x,y) dy$

## Acknowledgements

This applet makes use of the excellent d3js javascript library.

## Licence

Creative Commons Attribution 4.0 International. License (CC BY 4.0). To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.