The inverse transform method samples a continuous distribution by converting a standard continuous-uniform probability through the distribution’s quantile function. Begin with a normalized PDF, integrate it to obtain the nondecreasing CDF, then invert that CDF. section 15.4
The sampling pipeline is:
- Normalize the desired PDF so its total area is 1.0. section 15.4
- Integrate the PDF to obtain a CDF, which maps values to probabilities. section 15.4
- Use the CDF’s inverse, called the quantile function, to map a uniform probability to a sampled value. section 15.4
Sampling a Cauchy distribution
The standard Cauchy distribution supplies a concrete case: its PDF is 1/(π(1+x²)), its CDF is atan(x)/π + 0.5, and its quantile function is tan(π(p−0.5)). Feeding uniform probabilities into that quantile function produces samples with visibly heavier tails than normal samples. section 15.4
