FINITE QUEUING TABLES
CONTENTS CHAPTER CONCEPTS OF PROBABILITY INTRODUCTION BASIC CONCEPTS DISCRETE PROBABILITY DISTRIBUTIONS BERNOULLI DISTRIBUTION BINOMIAL DISTRIBUTION POISSON DISTRIBUTION CONTINUOUS PROBABILITY DISTRIBUTIONS PROBABILITY DISTRIBUTION FUNCTION, PROBABILITY DENSITY FUNCTION, AND CUMULATIVE DISTRIBUTION FUNCTION THE NORMAL DISTRIBUTION EXPONENTIAL DISTRIBUTION STUDENT’S T-DISTRIBUTION EXTREME VALUE DISTRIBUTION GENERALIZED EXTREME VALUE DISTRIBUTION STATISTICAL MOMENTS AND QUANTILES LOCATION DISPERSION ASYMMETRY CONCENTRATION IN TAILS STATISTICAL MOMENTS QUANTILES SAMPLE MOMENTS JOINT PROBABILITY DISTRIBUTIONS CONDITIONAL PROBABILITY DE?NITION OF JOINT PROBABILITY DISTRIBUTIONS MARGINAL DISTRIBUTIONS DEPENDENCE OF RANDOM VARIABLES COVARIANCE AND CORRELATION MULTIVARIATE NORMAL DISTRIBUTION ELLIPTICAL DISTRIBUTIONS COPULA FUNCTIONS PROBABILISTIC INEQUALITIES CHEBYSHEV’S INEQUALITY FR´ECHET-HOEFFDING INEQUALITY SUMMARY CHAPTER OPTIMIZATION UNCONSTRAINED OPTIMIZATION MINIMA AND MAXIMA OF A DIFFERENTIABLE FUNCTION CONVEX FUNCTIONS QUASICONVEX FUNCTIONS CONSTRAINED OPTIMIZATION LAGRANGE MULTIPLIERS CONVEX PROGRAMMING LINEAR PROGRAMMING QUADRATIC PROGRAMMING PROBABILITY METRICS MEASURING DISTANCES: THE DISCRETE CASE SETS OF CHARACTERISTICS DISTRIBUTION FUNCTIONS JOINT DISTRIBUTION PRIMARY, SIMPLE, AND COMPOUND METRICS AXIOMATIC CONSTRUCTION PRIMARY METRICS SIMPLE METRICS COMPOUND METRICS MINIMAL AND MAXIMAL METRICS TECHNICAL APPENDIX REMARKS ON THE AXIOMATIC CONSTRUCTION OF PROBABILITY METRICS EXAMPLES OF PROBABILITY DISTANCES MINIMAL AND MAXIMAL DISTANCES IDEAL PROBABILITY METRICS THE CLASSICAL CENTRAL LIMIT THEOREM THE BINOMIAL APPROXIMATION TO THE NORMAL DISTRIBUTION THE GENERAL CASE ESTIMATING THE DISTANCE FROM THE LIMIT DISTRIBUTION THE GENERALIZED CENTRAL LIMIT THEOREM STABLE DISTRIBUTIONS MODELING FINANCIAL ASSETS WITH STABLE DISTRIBUTIONS CONSTRUCTION OF IDEAL PROBABILITY METRICS DE?NITION EXAMPLES THE CLT CONDITIONS REMARKS ON IDEAL METRICS CHOICE UNDER UNCERTAINTY EXPECTED UTILITY THEORY ST PETERSBURG PARADOX THE VON NEUMANN–MORGENSTERN EXPECTED UTILITY THEORY TYPES OF UTILITY FUNCTIONS STOCHASTIC DOMINANCE FIRST-ORDER STOCHASTIC DOMINANCE SECOND-ORDER STOCHASTIC DOMINANCE ROTHSCHILD-STIGLITZ STOCHASTIC DOMINANCE THIRD-ORDER STOCHASTIC DOMINANCE EF?CIENT SETS AND THE PORTFOLIO CHOICE PROBLEM RETURN VERSUS PAYOFF PROBABILITY METRICS AND STOCHASTIC DOMINANCE THE AXIOMS OF CHOICE STOCHASTIC DOMINANCE RELATIONS OF ORDER N RETURN VERSUS PAYOFF AND STOCHASTIC DOMINANCE OTHER STOCHASTIC DOMINANCE RELATIONS RISK AND UNCERTAINTY MEASURES OF DISPERSION STANDARD DEVIATION MEAN ABSOLUTE DEVIATION SEMISTANDARD DEVIATION AXIOMATIC DESCRIPTION DEVIATION MEASURES PROBABILITY METRICS AND DISPERSION MEASURES MEASURES OF RISK VALUE-AT-RISK COMPUTING PORTFOLIO VAR IN PRACTICE BACKTESTING OF VAR COHERENT RISK MEASURES RISK MEASURES AND DISPERSION MEASURES RISK MEASURES AND STOCHASTIC ORDERS CONVEX RISK MEASURES PROBABILITY METRICS AND DEVIATION MEASURES AVERAGE VALUE-AT-RISK AVAR ESTIMATION FROM A SAMPLE COMPUTING PORTFOLIO AVAR IN PRACTICE THE MULTIVARIATE NORMAL ASSUMPTION THE HISTORICAL METHOD THE HYBRID METHOD THE MONTE CARLO METHOD BACKTESTING OF AVAR SPECTRAL RISK MEASURES RISK MEASURES AND PROBABILITY METRICS CHARACTERISTICS OF CONDITIONAL LOSS DISTRIBUTIONS HIGHER-ORDER AVAR THE MINIMIZATION FORMULA FOR AVAR AVAR FOR STABLE DISTRIBUTIONS ETL VERSUS AVAR REMARKS ON SPECTRAL RISK MEASURES OPTIMAL PORTFOLIOS MEAN-VARIANCE ANALYSIS MEAN-VARIANCE OPTIMIZATION PROBLEMS THE MEAN-VARIANCE EF?CIENT FRONTIER MEAN-VARIANCE ANALYSIS AND SSD ADDING A RISK-FREE ASSET MEAN-RISK ANALYSIS MEAN-RISK OPTIMIZATION PROBLEMS THE MEAN-RISK EF?CIENT FRONTIER MEAN-RISK ANALYSIS AND SSD RISK VERSUS DISPERSION MEASURES TYPES OF CONSTRAINTS QUADRATIC APPROXIMATIONS TO UTILITY FUNCTIONS SOLVING MEAN-VARIANCE PROBLEMS IN PRACTICE SOLVING MEAN-RISK PROBLEMS IN PRACTICE REWARD-RISK ANALYSIS BENCHMARK TRACKING PROBLEMS THE TRACKING ERROR PROBLEM RELATION TO PROBABILITY METRICS EXAMPLES OF R D METRICS NUMERICAL EXAMPLE DEVIATION MEASURES AND R D METRICS REMARKS ON THE AXIOMS MINIMAL R D METRICS LIMIT CASES OF L? P(X, Y) AND ? P(X, Y) COMPUTING R D METRICS IN PRACTICE PERFORMANCE MEASURES REWARD-TO-RISK RATIOS RR RATIOS AND THE EF?CIENT PORTFOLIOS LIMITATIONS IN THE APPLICATION OF REWARD-TO-RISK RATIOS THE STARR THE SORTINO RATIO THE SORTINO-SATCHELL RATIO A ONE-SIDED VARIABILITY RATIO THE RACHEV RATIO REWARD-TO-VARIABILITY RATIOS RV RATIOS AND THE EF?CIENT PORTFOLIOS THE SHARPE RATIO THE CAPITAL MARKET LINE AND THE SHARPE RATIO EXTENSIONS OF STARR QUASICONCAVE PERFORMANCE MEASURES THE CAPITAL MARKET LINE AND QUASICONCAVE RATIOS NONQUASICONCAVE PERFORMANCE MEASURES PROBABILITY METRICS AND PERFORMANCE MEASURES RANDOM VARIABLES CAN BE CLASSI?ED AS EITHER DISCRETE OR CONTINUOUS WE BEGIN WITH DISCRETE PROBABILITY DISTRIBUTIONS AND THEN PROCEED TO CONTINUOUS PROBABILITY DISTRIBUTIONS 1THE PRECISE MATHEMATICAL DE?NITION IS THAT A RANDOM VARIABLE IS A MEASURABLE FUNCTION FROM A PROBABILITY SPACE INTO THE SET OF REAL NUMBERS IN THIS CHAPTER, THE READER WILL REPEATEDLY BE CONFRONTED WITH IMPRECISE DE?NITIONS THE AUTHORS HAVE INTENTIONALLY CHOSEN THIS WAY FOR A BETTER GENERAL UNDERSTANDABILITY AND FOR THE SAKE OF AN INTUITIVE AND ILLUSTRATIVE DESCRIPTION OF THE MAIN CONCEPTS OF PROBABILITY THEORY IN ORDER TO INFORM ABOUT EVERY OCCURRENCE OF LOOSENESS AND LACK OF MATHEMATICAL RIGOR, WE HAVE FURNISHED MOST IMPRECISE DE?NITIONS WITH A FOOTNOTE GIVING A REFERENCE TO THE EXACT DE?NITION 2FOR MORE DETAILED AND/OR COMPLEMENTARY INFORMATION, THE READER IS REFERRED TO THE TEXTBOOKS OF LARSEN AND MARX (1986), SHIRYAEV (1996), AND BILLINGSLEY (1995) DISCRETE PROBABILITY DISTRIBUTIONS AS THE NAME INDICATES, A DISCRETE RANDOM VARIABLE LIMITS THE OUTCOMES WHERE THE VARIABLE CAN ONLY TAKE ON DISCRETE VALUES FOR EXAMPLE, CONSIDER THE DEFAULT OF A CORPORATION ON ITS DEBT OBLIGATIONS OVER THE NEXT ?VE YEARS THIS RANDOM VARIABLE HAS ONLY TWO POSSIBLE OUTCOMES: DEFAULT OR NONDEFAULT HENCE, IT IS A DISCRETE RANDOM VARIABLE CONSIDER AN OPTION CONTRACT WHERE FOR AN UPFRONT PAYMENT (I E , THE OPTION PRICE) OF $50,000, THE BUYER OF THE CONTRACT RECEIVES THE PAYMENT GIVEN IN TABLE 1 1 FROM THE SELLER OF THE OPTION DEPENDING ON THE RETURN ON THE S&P 500 INDEX IN THIS CASE, THE RANDOM VARIABLE IS A DISCRETE RANDOM VARIABLE BUT ON THE LIMITED NUMBER OF OUTCOMES 3PRECISELY, ONLY CERTAIN SUBSETS OF THE SAMPLE SPACE ARE CALLED EVENTS IN THE CASE THAT THE SAMPLE SPACE IS REPRESENTED BY A SUBINTERVAL OF THE REAL NUMBERS, THE EVENTS CONSIST OF THE SO-CALLED ‘‘BOREL SETS ’’ FOR ALL PRACTICAL APPLICATIONS, WE CAN THINK OF BOREL SETS AS CONTAINING ALL SUBSETS OF THE SAMPLE SPACE IN THIS CASE, THE SAMPLE SPACE TOGETHER WITH THE SET OF EVENTS IS DENOTED BY (R, B) SHIRYAEV (1996) PROVIDES A PRECISE DE?NITION 4PROBABILITY IS VIEWED AS A FUNCTION ENDOWED WITH CERTAIN PROPERTIES, TAKING EVENTS AS AN ARGUMENT AND PROVIDING THEIR PROBABILITIES AS A RESULT THUS, ACCORDING TO THE MATHEMATICAL CONSTRUCTION, PROBABILITY IS DE?NED ON THE ELEMENTS OF THE SET A (CALLED SIGMA-?ELD OR SIGMA-ALGEBRA) TAKING VALUES IN THE INTERVAL [0, 1], P : A ?[0, 1] FINITE QUEUING TABLES CONCEPTS OF PROBABILITY THE PROBABILISTIC TREATMENT OF DISCRETE RANDOM VARIABLES IS COMPARA- TIVELY EASY: ONCE A PROBABILITY IS ASSIGNED TO ALL DIFFERENT OUTCOMES, THE PROBABILITY OF AN ARBITRARY EVENT CAN BE CALCULATED BY SIMPLY ADDING THE SINGLE PROBABILITIES IMAGINE THAT IN THE ABOVE EXAMPLE ON THE S&P