View source for Sampling from a probability density function



Frequently, a random sample is required that exhibits a known [[Probability density function#217|probability density function]]. Random number generators, on the other hand, usually supply samples with a uniform distribution, typically between 0 and 1. What is needed is a recipe for converting the flat probability density function (pdf) to the desired (``target'') pdf. Call the target pdf ''f''(''x''); [[Distribution#65|distribution]] function is defined by

[[File:hepa_img985.gif|316x36px]]

By definition,

[[File:hepa_img986.gif|315x39px]]

''F''(''x'') is uniformly distributed over the interval [0,1], for all ''f''(''x''); hence a uniformly distributed random number ''r'' can be interpreted as a random value of ''F''(''x''). If a generator supplies ''r'', the variable ''x'' is obtained by solving

[[File:hepa_img987.gif|307x38px]]

for ''x''. If ''f''(''x'') is not analytically integrable, or only known in discretized form (e.g. as a histogram), simple methods of numerical integration and, possibly, interpolation will usually suffice.

[[Category:W.Krisher and R.Bock]]
[[Category:Data analysis]]
[[Category:Statistics]]

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