Existence of Moments in Distributions of the Form Tan(X) | Chapter 4 | Mathematics and Computer Science: Contemporary Developments Vol. 5

In this work, we consider the existence of the moments of functions of random variables supported on a bounded interval. Our approach begins by working with an arbitrary diffeomorphism, but later we restrict attention to the tan function–the corresponding distribution is a generalization of the Cauchy distribution, which is derived when one applies tan to a uniformly distributed variable. For a continuous random variable X, we derive a necessary and sufficient condition for the existence of a moment of a given order of the distribution of tan(X) in terms of the behaviour of the probability density of X near the points ± \(\frac{\pi}{2}\). As a consequence, we obtain classes of examples, somewhere the moments exist and somewhere they do not at all.

 

Author (s) Details

Peter Kopanov
Department of Mathematics and Informatics, Plovdiv University 'Paisii Hilendarski', 4000, Plovdiv, Bulgaria.

 

Miroslav Marinov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.

 

Atakan Salimov
Technical University of Sofia, Faculty of Computer Systems and Technologies, Bulgaria.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v5/2243