About

Dr. Sheldon Mark Ross is the Daniel J. Epstein Chair and Professor of Industrial and Systems Engineering at the University of Southern California Viterbi School of Engineering. He holds a B.S. degree in Mathematics from Brooklyn College, obtained in 1963, an M.S. degree in Mathematics from Purdue University in 1964, and a Ph.D. in Statistics from Stanford University in 1968. Prior to joining USC in 2004, he served as a Professor at the University of California, Berkeley from 1976. His research focuses on applied probability models, financial engineering, simulation, and stochastic dynamic programming. Dr. Ross is actively involved in scholarly publishing, serving as the Editor for Probability in the Engineering and Informational Sciences, the Advisory Editor for the International Journal of Quality Technology and Quantitative Management, and an Editorial Board Member of the Journal of Bond Trading and Management. He has received recognition for his contributions to the field, including being named a Fellow of INFORMS in 2013.

Research topics

• Statistics
• Artificial Intelligence
• Computer Science
• Mathematics
• Discrete mathematics
• Finance
• Applied mathematics
• Economics
• Mathematical economics

Selected publications

• Finding a good normal population

  Operations Research Letters · 2025-04-02

  article1st author
  PublisherDOI

• The time until a random walk exceeds a square root and other barriers

  Probability in the Engineering and Informational Sciences · 2025-09-03

  articleOpen access1st author

  Abstract This paper investigates the time N until a random walk first exceeds some specified barrier. Letting $X_i, i \geq 1,$ be a sequence of independent, identically distributed random variables with a log-concave density or probability mass function, we derive both lower and upper bounds on the probability $P(N \gt n),$ as well as bounds on the expected value $E[N].$ On barriers of the form $a + b \sqrt{k},$ where a is nonnegative, b is positive, and k is the number of steps, we provide additional bounds on $E[N].$

  PublisherOA PDFDOI

• Improved bounds on the probability of a union and on the number of events that occur

  Operations Research Letters · 2025-07-29

  articleOpen accessSenior author

  Let A 1 , A 2 , … , A n be events in a sample space. Given the probability of the intersection of each collection of up to k + 1 of these events, what can we say about the probability that at least r of the events occur? This question dates back to Boole in the 19th century, and it is well known that the odd partial sums of the Inclusion-Exclusion formula provide upper bounds, while the even partial sums provide lower bounds. We give a combinatorial characterization of the error in these bounds and use it to derive a very simple proof of the strongest possible bounds of a certain form, as well as a couple of improved bounds. The new bounds use more information than the classical Bonferroni-type inequalities, and are often sharper.

  PublisherDOI

• First ahead by at least k multinomial game

  Annals of Operations Research · 2025-04-07

  articleOpen access1st authorCorresponding

  Abstract Multinomial trials having probabilities $$p_i = \frac{v_i}{ \sum _{j=1}^n v_j}, \, i = 1, \ldots , n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mfrac> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mrow> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> are observed until one of the outcomes, called the winning outcome, has occurred at least k more times than each of the others. We give an efficient simulation approach for estimating the probability that each outcome is the winner as well as the mean number of trials needed. We also show that the probability that outcome i wins is an increasing function of $$v_i,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and is greater than the probability that outcome j wins when $$v_i &gt; v_j.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>

  PublisherOA PDFDOI

• Improved Bounds on the Probability of a Union and on the Number of Events that Occur

  ArXiv.org · 2025-05-18

  preprintOpen accessSenior author

  Let $A_1, A_2, \ldots, A_n$ be events in a sample space. Given the probability of the intersection of each collection of up to $k+1$ of these events, what can we say about the probability that at least $r$ of the events occur? This question dates back to Boole in the 19th century, and it is well known that the odd partial sums of the Inclusion- Exclusion formula provide upper bounds, while the even partial sums provide lower bounds. We give a combinatorial characterization of the error in these bounds and use it to derive a very simple proof of the strongest possible bounds of a certain form, as well as a couple of improved bounds. The new bounds use more information than the classical Bonferroni-type inequalities, and are often sharper.

  PublisherOA PDFDOI

• A Second Course in Probability

  2023 · 23 citations

  1st authorCorresponding
  • Computer Science
  • Artificial Intelligence
  • Mathematical economics

  Written by Sheldon Ross and Erol Peköz, this text familiarises you with advanced topics in probability while keeping the mathematical prerequisites to a minimum. Topics covered include measure theory, limit theorems, bounding probabilities and expectations, coupling and Stein's method, martingales, Markov chains, renewal theory, and Brownian motion. No other text covers all these topics rigorously but at such an accessible level - all you need is an undergraduate-level understanding of calculus and probability. New to this edition are sections on the gambler's ruin problem, Stein's method as applied to exponential approximations, and applications of the martingale stopping theorem. Extra end-of-chapter exercises have also been added, with selected solutions available.This is an ideal textbook for students taking an advanced undergraduate or graduate course in probability. It also represents a useful resource for professionals in relevant application domains, from finance to machine learning.

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• Brownian Motion

  Cambridge University Press eBooks · 2023-08-31

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• Measure Theory and Laws of Large Numbers

  Cambridge University Press eBooks · 2023-08-31

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• Conditional Expectation and Martingales

  Cambridge University Press eBooks · 2023-08-31

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• The Exponential Distribution and the Poisson Process

  Elsevier eBooks · 2023-07-14

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Recent grants

• Models for Choosing the Best Population

  NSF · $423k · 2022–2027

• Stochastic Sequential Assignment Problems

  NSF · $248k · 2012–2017

• Collaborative Research: Theoretical and Algorithmic Advances in Sequential Adaptive Decisions

  NSF · $350k · 2017–2021

Frequent coauthors

• Erol A. Peköz

  Boston University

  46 shared

• Samim Ghamami

  University of California, Berkeley

  21 shared

• Cyrus Derman

  19 shared

• Mark Brown

  Columbia University

  18 shared

• Ilan Adler

  Healthcentric Advisors

  14 shared

• Sridhar Seshadri

  University of Illinois Urbana-Champaign

  13 shared

• Richard M. Karp

  11 shared

• G. J. Lieberman

  10 shared

Education

• Ph.D., Industrial and Systems Engineering

  University of Southern California

  1990

• M.S., Industrial and Systems Engineering

  University of Southern California

  1986

• B.S., Industrial and Systems Engineering

  University of Southern California

  1984

Awards & honors

• INFORMS -- The Institute for Operations Research and the Man…