DCCN'2025

Keynote

THE ALMOST LACK OF MEMORY (ALM) DISTRIBUTIONS AND THEIR APPLICATIONS

Prof. Boyan Dimitrov(USA)

Long ago baron Augustin-Louis Cauchy (1789 - 1857) proved in 1821 that if the functional equation f(x+y)=f(x)f(y)f(x+y)=f(x)f(y) holds for any non-negative arguments xx and yy, then the function f(x)f(x) exponential f(x)=ecxf(x)=e^{cx} function. When applied to the probability property of a random lifetime of a technical unit XX it looks P{X≥x+y}=P{X≥x}P{X≥y}P\{X \ge x+y\}= P \{X \ge x\}P\{X \ge y\}. From Kaushy theorem and Probability it follows that the lifetime probability distribution function has the form F{X≥′x}=1−e−axF \{X \ge' x\}= 1 - e^{-ax}. And it follows that if this unit still works (is alive at age yy), the chances to stay alive some more time x, is the same as when just starts functioning: P{X≥x+y∣X≥y}=P{X≥x}P\{X \ge x+y | X \ge y \}= P\{X \ge x\}.

A conventional reading of this property is known as Lack-of-Memory (LM) property at any age yy. It means that at any age y the units with exponentially distributed lifetimes lose the memory about their current age and behave as a just newborn. This is a characteristic property that helps in practice to recognize the lifetime distribution of technical items.

In a series of works with numerous colleagues (please, see the references) on similar properties that may be used in practice to recognize the lifetime distribution of technical items. And we found that if a lifetime shows the lack of memory at a given age c.

It will lose the memory at any age mc integer multiple to the constant c for infinitely many times m=2,3,4,…m=2,3, 4,…. For this reason, we named these distributions ALM distributions. And we found the mathematical form of this class of probability distributions, established numerous mathematical presentations, physical properties, and found various practical applications.

REFERENCES

A lectrure "Long life with Mathematics" on the topic of this talk, presented at Kettering university, is available at the link below.

https://www.youtube.com/watch?v=2GouA8Y1oJA

About the speaker

Boyan Dimitrov avatar

Prof. Boyan Dimitrov

USA
  • Kettering University
    Professor

Degree Information

  1. Dr. of Mathematical Sciences, Probability and Statistics, - Faculty of Mathematics and Computer Science, Sofia University, Sofia, Bulgaria. Dissertation: "Mathematical Methods for investigations of queuing systems", May 1986, (in Bulgarian).

  2. Ph. D. Physics & Mathematics, Probability and Statistics, Mechmath of the Moscow State Univ. Michael Lomonosov. Dissertation: "Some applications of the random sums method" (in Russian), June 1971, Supervisor: Prof. B. V. Gnedenko.

  3. M.A. Mathematical Education with specialization in Probability and Statistics, Faculty of Mathematics and Computer Science, Sofia University, Sofia, Bulgaria. July, 1966. Thesis: "Limit theorems for infinitesimal renewal processes" (in Bulgarian).

Employment record

  1. July 1995 - June 2001: Associate Professor of Mathematics (Probability and Statistics), GMI Engineering and Management Inst. (now Kettering)
  2. October 1993 - June 1995: Adjunct Professor of Mathematics (Probability and Statistics), GMI Engineering and Management Inst.
  3. January - June 1993: Visiting Professor at Concordia University, Montreal, Canada, Dept. of Mathematics and Statistics.
  4. September - December 1992: Visiting Professor at McGill University, Dept. of Mathematics and Statistics, Montreal, Canada and Researcher at Concordia University.
  5. January - August 1992: Visiting Professor at Concordia University, Montreal, Canada, Dept. of Mathematics and Statistics.
  6. January 1989 - December 1991: Professor of Probability and Statistics, Faculty of Mathematics and Informatics, University of Sofia, Bulgaria.
  7. September - December 1988: Visiting professor at Concordia University, Montreal, Canada, Dept. of Mathematics and Statistics.
  8. December 1976 - August 1988: Associate professor, Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria.
  9. August 1971 - December 1976: Researcher/Lecturer in Mathematics; Center of Mathematics and Mechanics, Sofia, Bulgaria.
  10. October 1968 - July 1971: Post-graduate (PhD) student, Moscow State University, Moscow, Russia.
  11. September 1966 - October 1968: Researcher/Lecturer in Mathematics; Center of Mathematics and Mechanics, Sofia, Bulgaria.