page_title: Random-fuzzy variable
section_title: The random distribution(rrandom)
hierarchical_section_title: Random-fuzzy variable / Construction / The random distribution(rrandom)
caption_attribution_description: English: A uniform probability distribution and its equivalent distribution in possibility domain done using probability possibility transformation.
context_page_description: In measurements, the measurement obtained can suffer from two types of uncertainties. The first is the random uncertainty which is due to the noise in the process and the measurement. The second contribution is due to the systematic uncertainty which may be present in the measuring instrument. Systematic errors, if detected, can be easily compensated as they are usually constant throughout the measurement process as long as the measuring instrument and the measurement process are not changed. But it can not be accurately known while using the instrument if there is a systematic error and if there is, how much? Hence, systematic uncertainty could be considered as a contribution of a fuzzy nature.
  This systematic error can be approximately modeled based on our past data about the measuring instrument and the process.
  Statistical methods can be used to calculate the total uncertainty from both systematic and random contributions in a measurement. But, the computational complexity is very high and hence, are not desirable.
  L.A.Zadeh introduced the concepts of fuzzy variables and fuzzy sets. Fuzzy variables are based on the theory of possibility and hence are possibility distributions.
context_section_description: rrandom is the possibility distribution of the random contributions to the uncertainty. Any measurement instrument or process suffers from random error contributions due to intrinsic noise or other effects.
  This is completely random in nature and is a normal probability distribution when several random contributions are combined according to the Central limit theorem.
  But, there can also be random contributions from other probability distributions such as a uniform distribution, gamma distribution and so on.
  The probability distribution can be modeled from the measurement data. Then, the probability distribution can be used to model an equivalent possibility distribution using the maximally specific probability-possibility transformation.
  Some common probability distributions and the corresponding possibility distributions can be seen in the figures.
caption: Uniform distribution in probability and possibility.
exif: {}