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.
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 \(X\).\[p(x) = \int p(x,y) dy\]
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