Abstract
In this article we are going to discuss imprecise randomness using the mathematics of partial presence. The mathematical explanations of imprecise randomness would actually be complete only if it is explained with reference to the Randomness-Impreciseness Consistency Principle. In this article, we have described imprecise randomness with reference to a numerical example of the two sample t-test.
Highlights
If the realizations of a random variable are imprecise in the sense that two independent laws of randomness can define the presence level of values of the variable in a given interval, we would have to deal with the matters using the idea of imprecise randomness (Baruah (2012))
When a non-rejectable hypothesis is made imprecise, there may still be a probability that the imprecise hypothesis would be found rejectable, the probability of rejection decided by the right reference function
If a rejectable hypothesis is made imprecise, there may still be a probability that the imprecise hypothesis would be found nonrejectable, the probability of non rejection being decided by the left reference function this time
Summary
If the realizations of a random variable are imprecise in the sense that two independent laws of randomness can define the presence level of values of the variable in a given interval, we would have to deal with the matters using the idea of imprecise randomness (Baruah (2012)). In the Zadehian definition of complementation, fuzzy membership function and fuzzy membership value have been taken to be the same, and that is where the problem lies. Fuzzy membership function and fuzzy membership value are two different things for the complement of a normal fuzzy set (Baruah (1999, 2011)). Our approach is different from their’s in the sense that we would be defining imprecise randomness using the Randomness- Impreciseness Consistency Principle together with our definition of complement of an imprecise set. Baruah has already established(Baruah (2012)) that every law of impreciseness can be expressed in terms of two laws of randomness, with randomness defined in the measure theoretic sense. We shall first discuss about Baruah’s RandomnessImpreciseness Consistency Principle andthe complement of an imprecise set. Thereafter we shall discuss the matters with reference to testing an imprecise hypothesis with reference to thetwo sample t- test
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