The Theory of Probability

by Boris V. Gnedenko, Борис Владимирович Гнеденко

This classic book is intended to be the first introduction to probability and statistics written with an emphasis on the analytic approach to the problems discussed. Topics include the axiomatic setup of probability theory, polynomial distribution, finite Markov chains, distribution functions and convolution, the laws of large numbers (weak and strong), characteristic functions, the central limit theorem, infinitely divisible distributions, and Markov processes. Written in a clear and concise style, this book by Gnedenko can serve as a textbook for undergraduate and graduate courses in probability.

Genres: Mathematics

529 pages, Hardcover

First published April 1, 1970

Reviews

[Douglas · Rating 5 out of 5]

This is a preliminary review. First, B. V. Gnedenko is a hero of mine, having opposed the Soviet regime and put himself at great personal risk on that account. According to one short biography Gnedenko was,

"...denounced by one of the members of the [Mathematics Institute of Moscow State University]...[and]...was imprisoned with 120 other prisoners in a cell built for six people and was constantly interrogated about statements he had made on the summer trip. His interrogators demanded that he...confirm that Kolmogorov was the ringleader of a group of "enemies of the people" centred in the mathematics department. Though interrogated daily over a six-month period, held in grim condition, and promised his release if he cooperated, he refused to admit even the possibility of such an interpretation, knowing that there could be no hard evidence, and that the fate of all, himself included, depended on his resolution."

In other words, Gnedenko wouldn't rat out his colleagues--in this case Kolmogorov. Okay!, so this guy is no ordinary mathematician of the white wine and brie sort; the types found in nice little college towns such as Ann Arbor, Madison or Berkeley. This suggests an interesting perspective that might flow over into his view of mathematics and, specifically probability theory. I believe this is the case and the reader will learn for himself.

Gnedenko was committed to teaching mathematics and writing popular and semi-popular renditions of mathematical topics. The Theory of Probability is an obvious product of his commitment and, as has been noted by a reviewer at Amazon, is suitable for independent study.

Gnedenko first introduced me to the notion that all which is random is not necessarily probability. That is, a random event may only be an event whose outcome, given a set of conditions, is not determinate; one cannot say certainty whether, under the specified conditions, it will occur or not. But the generality Gnedenko's notion of randomness is not sufficient to assign a probability to the event. For that to be the case one must specify a "sufficiently definite set of external conditions". Gnedenko seems to have an experimental viewpoint in mind as he mainly cites examples from physical science. It is pure mishmash to demand a probability for an event under any set of conditions. This leads me to wonder whether, outside physical science--to include the laws or ordinary gambling--it is possible to establish sufficiently specific conditions? I know this is a pretty sweeping question but I have in mind disciplines like economics, sociology and psychology where probability theory has been increasingly invoked in recent decades. The idea that mass behavior is related to what Gnedenko calls mass events has been debated ever since the first applications of probability to these areas.

All physical laws require, what are sometimes called constitutive relationships. That is, there must be some way to determine the constants which occur in the relationship. One cannot say that all the symbols occurring in a mathematical formula are "variables". Without these constants or relationships tantamount to constants, there is too much generality and meaningful solutions cannot be found. I'm thinking at present of Maxwell's equations, for example. And, in fact, it is the constitutive relationships which tie the equations to matter and reduce the generality of the mathematical relationships.

It has been argued by many that the (meaningful) application of mathematics to economics, for example, is hampered because there exist no real constants. All of the symbols in a mathematical expression in economics are variables. For example, a quick web search turned up this,

New York University economist Paul Romer -- hardly a lightweight when it comes to equations -- recently complained about how economists use math as a tool of rhetoric instead of a tool to understand the world. http://www.bloombergview.com/articles...

Reading on in the article we find that economics tends the put evidence--read results from experiments--behind theory. Now, a mathematical expression in a theory without evidence contains only variables (excluding, naturally, purely mathematical constants such as pi and e). This reversal of evidence and theory is not surprising because, when it comes down to cases, what is an economics experiment? A social experiment to say the least, and that raises a nest of problems outside of science.

Generally, the use of mathematics in the behavioral sciences must be accomplished by minimizing or explaining away volition. And this is fatal to a science which claims to have something meaningful to say about volitional creatures i.e., men. Physics, on the other hand, excludes volition altogether. The earth revolves around the sun and it is possible to summarize this geometrically precisely because both bodies lack volition; though Dirac once complained to Heisenberg that at the sub-atomic or quantum level, it would seem that volition is almost the only thing left which could give continuity to events.