A continuous distribution models a random real number whose possible values occupy a continuous range rather than an enumerable set. A manufacturing process might produce part diameters with small, random variation; each sample is one real-valued measurement. section 15.1
A probability density function (PDF) assigns a non-negative weight to each real value. Probability over an interval comes from the area under the PDF there, divided by the total area. A PDF with total area 1 is normalized; many useful PDFs are not. section 15.1 section 15.6
Several operations connect the density to sampling:
- Integrate the PDF to obtain the cumulative distribution function (CDF), which gives the probability that a sample is less than a chosen value. section 15.6
- Invert the CDF to obtain the quantile function; applying it to a standard uniform sample produces a sample from the target distribution. section 15.6
- Use rejection sampling when the PDF is known but its quantile function is unavailable; a helper distribution supplies candidate points, though finding one can be difficult and many candidates may be rejected. section 15.6
Uniform samples and a visible sanity check
The continuous uniform distribution makes every value between a minimum and maximum equally likely. In C#, a standard instance can generate values with `Random.NextDouble()`. Drawing 100,000 samples and placing them into 40 histogram columns should produce columns with roughly equal heights, while random variation prevents a perfectly flat result. section 15.2
double P(double x) =>
0.0 <= x && x <= 1.0 ?
Math.Pow(x, 0.5) * Math.Pow(1.0 - x, 1.5) :
0.0;
Inspect a concrete beta-distribution PDF: it is nonzero only from 0 to 1 and assigns varying density across that interval. section 15.1
