• Naresh Kumar
• Dr. Gaurav Varshney

We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line [0, ∞). These equations arise when studying distributional equations of the type Z d= X + TZ, where the random variable T ≥ 0 has known distribution, while the distribution of the random variable Z ≥ 0 is a transformation of that of X ≥ 0, and we want to find the distribution of X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results that are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.

distributional equation; power-mixture transform; functional equation; characterization of distributions

"MEAN AND VARIANCE BOUNDS CHARACTERIZATION OF PROBABILITY DISTRIBUTIONS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.6, Issue 4, page no.246-255, April-2019, Available :http://www.jetir.org/papers/JETIR1904U87.pdf

"MEAN AND VARIANCE BOUNDS CHARACTERIZATION OF PROBABILITY DISTRIBUTIONS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.6, Issue 4, page no. pp246-255, April-2019, Available at : http://www.jetir.org/papers/JETIR1904U87.pdf