Is there an absolutely continuous random variable with equal probability density and cumulative distribution functions in its support? Is it unique? What about in the discrete case?
Abstract:This paper inquires about the existence and uniqueness of a univariate continuous random variable for which both cumulative distribution and density functions are equal and asks about the conditions under which a possible extrapolation of the solution to the discrete case is possible. The issue is presented and solved as a problem and allows to obtain a new family of probability distributions. The different approaches followed to reach the solution could also serve to warn about some properties of density and … Show more
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