[Search][Subject Index][MathMap][Tour][Help!]

	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 60: Probability theory and stochastic processes

Introduction

Probability theory is simply enumerative combinatorial analysis when applied to finite sets; thus the techniques and results resemble those of discrete mathematics. The theory comes into its own when considering infinite sets of possible outcomes. This requires much measure theory (and a careful interpretation of results!) More analysis enters with the study of distribution functions, and limit theorems implying central tendencies. Applications to repeated transitions or transitions over time lead to Markov processes and stochastic processes. Probability concepts are applied across mathematics when considering random structures, and in particular lead to good algorithms in some settings even in pure mathematics.

History

Applications and related fields

Some material in probability (especially foundational questions) is really measure theory. The topic of randomly generating points on a sphere is included here but there is another page with general discussions of spheres. Probability questions given a finite sample space are usually "just" a lot of counting, and so are included with combinatorics.

For additional applications, See 11KXX, 62-XX, 90-XX, 92-XX, 93-XX, 94-XX. For numerical results, See 65U05

Subfields

• 60A: Foundations of probability theory
• 60B: Probability theory on algebraic and topological structures
• 60C05: Combinatorial probability
• 60D05: Geometric probability, stochastic geometry, random sets, See also 52A22, 53C65
• 60E: Distribution theory, see also 62EXX, 62HXX
• 60F: Limit theorems, see also 28DXX, 60B12
• 60G: Stochastic processes
• 60H: Stochastic analysis, see also 58G32
• 60J: Markov processes
• 60K: Special processes

This is one of the larger areas in the Math Reviews database. (The subfield 60K25, Queueing theory, is among the largest of the five-digit subfields.)

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Bosch, A. J.: "What is a random variable?" (Dutch) Nieuw Tijdschr. Wisk. 64 (1976/77), no. 5, 241--250. MR58#2932

Doob, J. L.: "What is a martingale?", Amer. Math. Monthly 78 (1971) 451--463. MR44#1094

Doob, J. L.: "What is a stochastic process?", Amer. Math. Monthly 49 (1942) 648--653. MR4,103b

Resnick, Sidney: "Adventures in stochastic processes", Birkhäuser Boston, Inc., Boston, MA, 1992. 626 pp. ISBN 0-8176-3591-2 MR93m:60004

Ocone, Daniel L.: "A guide to the stochastic calculus of variations", Stochastic analysis and related topics (Silivri, 1986), 1--79, Lecture Notes in Math., 1316; Springer, Berlin-New York, 1988. MR89h:60093

Wise, Gary L.; Hall, Eric B.: "Counterexamples in probability and real analysis", Oxford University Press, New York, 1993, ISBN 0-195-07068-2

Stoyanov, Jordan M., "Counterexamples in probability", Wiley, Chichester-New York, 1987, ISBN 0-471-91649-8

Software and tables

There is a node in the GAMS software tree for Simulation, stochastic modeling.

There is a newsgroup alt.sci.math.probability .

Other web sites with this focus

• Probability Abstract Database (search engine)
• Probability web
• UTK archives page
• Here are the AMS and Goettingen resource pages for area 60.

Selected topics at this site

• How to generate numbers with a Gaussian distribution (not a uniform one)?
• What is the Poisson distribution? (Analyzing coincidences of infrequent events)
• Randomly generating numbers to fit a specified distribution.
• Randomly generating numbers to fit a specified distribution.
• Randomly generating numbers to fit a specified distribution.
• How to generate a random variable with a given pdf
• Joint distribution for Brownian motion.
• How many shuffles before a deck of cards is "random"?
• Measuring the randomness of card shuffling with group theory.
• What are the odds in blackjack?
• Given a random ordering of k black balls and n-k white balls, what's the expected value for the length of the largest interval of black balls?
• The Hewitt-Savage 0-1 Law of random walks on the real line.
• Random walks on the plane and in R^n.
• Random walks on the sphere.
• Typical (but convoluted) counting problem.
• PDF for taxicab distances between two points in a rectangle.
• Citations for the Monty Hall problem.
• Strong vs. weak Law of Large Numbers.
• What do we learn from the law of large numbers?
• If shown one real number out of two, how can you guess whether it's the larger? (heh heh)
• Buffon's needle problem.
• Calculation of the expected number of pin-line crossings in the Buffon needle crossing problem.
• Pointers to the Buffon needle problem and experimental evaluation of Pi
• Cells either die or split in two; what's the long-term outcome?
• Pointer to a website simulating that Monty Hall paradox!
• Choose elements of a finite set without replacement. Probability of missing a particular one?
• Frequencies of patterns in cointosses [Denis Constales]
• What does it mean to select a random triangle? [Terry Moore]
• How to randomly generate points on an ellipse (ellipsoid)?
• Summary of methods for generating uniformly-distributed random points on a sphere [Dave Seaman]. (See also the sphere FAQ.)
• Book citation on the statistical analysis of spherical data.
• Probability that N randomly-selected points on a sphere lie in a single hemisphere.
• Proving the central limit theorem.
• Pointers regarding stochastic differential equations.
• Pointers for information on branching processes.
• The Secretary Problem (or, "How can a bachelor select a best wife?"): deciding when the best-so-far is nearly-best.
• Elementary description of the "Law of Large Numbers"
• Comparison of convergence in probability vs. convergence with probability one?
• Sample counterintuitive geometric probability question
• Limiting approximation of elementary expected value
• Chebyshev, Camp-Meidell estimates that a random variable lies near the mean
• With an ordered set of n random variables, probability that all initial sums are less than expected value (about 1/sqrt(n) )
• Why are Gaussian normal (or Poisson) distributions used in practice?
• Proof that a Poisson distribution is a limit of Bernoulli distributions?
• Marcinkiewicz' theorem: a probability distribution with only a finite number of non-vanishing cumulants must be Gaussian
• Probability an event occurs (or not) in a Poisson process
• Probability distribution of waiting time until n-th success: negative binomial (Pascal) distribution
• Comparison of negative binomial and hypergeometric distributions
• Simple example of hypergeometric distribution
• Use of a probability generating function to determine distribution of a sequential process
• Add random digits until a 0 appears; distribution of sums? (probability generating function)
• The sum of n independent, identically distributed exponentials
• Closed formula for position probabilities in random walk on a 2D square lattice
• Two different types of random walks on Z^n
• Probability of return in random walk on Z^n
• Asymptotics of random walks on finitely generated groups
• Expected return in martingales is zero (e.g. the stock market)
• Ito's formula, derivatives in stochastic processes
• Solution of sample stochastic differential equation
• Some good(?) random number generators, with C code, comparisons
• Weaknesses of Linear Congruential Generators as random number generators
• Pointer: codes for random number generation
• Generating random variables with a given correlation
• Generating random points in multidimensional polytopes
• Some thinking about a fair 3-sided coin
• Extreme-value distributions (invariant under MAX of two random variables)
• Distribution of sum (etc.) of two independent random variables
• Computing distribution of an algebraic combination of several independent random variables
• Bounding the variance of max(X,Y)
• Distribution of logarithm of a Poisson variable
• Computing Gaussian moments
• Modeling modem pool usage as an M/M/K/K system