The zeros of random polynomials cluster uniformly near the unit circle
... the unit circle is universal, in the sense that no independence or equidistribution on the coefficients is required, but only conditions on their size. Our method, based on elementary complex analysis, reduces both convergences (3) and (4) to the same problem, namely showing that LN (P ), defined in (5 ...
... the unit circle is universal, in the sense that no independence or equidistribution on the coefficients is required, but only conditions on their size. Our method, based on elementary complex analysis, reduces both convergences (3) and (4) to the same problem, namely showing that LN (P ), defined in (5 ...
The Probability that a Random - American Mathematical Society
... Every odd prime must pass this test. Moreover, Monier [3] and Rabin [4] have shown that if n > 1 is an odd composite, then the probability that it is a strong probable prime to a random base 6, 1 < 6 < n — 1, is less than 4. Let Pi{x) denote the same probability as P(x), except that (iii) is changed ...
... Every odd prime must pass this test. Moreover, Monier [3] and Rabin [4] have shown that if n > 1 is an odd composite, then the probability that it is a strong probable prime to a random base 6, 1 < 6 < n — 1, is less than 4. Let Pi{x) denote the same probability as P(x), except that (iii) is changed ...
A mini course on percolation theory
... Before starting the proof, we tell (or remind) the reader of another 0-1 Law which is different from Kolmogorov’s theorem and whose proof we will not give. I will not state it in its full generality but only in the context of percolation. (For people who are familiar with ergodic theory, this is not ...
... Before starting the proof, we tell (or remind) the reader of another 0-1 Law which is different from Kolmogorov’s theorem and whose proof we will not give. I will not state it in its full generality but only in the context of percolation. (For people who are familiar with ergodic theory, this is not ...
FACULTAD DE CIENCIAS EMPRESARIALES Y ECONOMIA Serie
... From Theorem 1, we can infer overcon…dence if a su¢ ciently large fraction of people (variable x in the theorem) believe su¢ ciently strongly (variable q) that they rank su¢ ciently high (variable y). From Theorem 2, we can infer overcon…dence if too few people who rank themselves high actually plac ...
... From Theorem 1, we can infer overcon…dence if a su¢ ciently large fraction of people (variable x in the theorem) believe su¢ ciently strongly (variable q) that they rank su¢ ciently high (variable y). From Theorem 2, we can infer overcon…dence if too few people who rank themselves high actually plac ...
1 1. Justification of analogical reasoning • an argument that it is
... “The supposition must be that m is an effect really dependent on some property of A, but we know not on which.” Derivative and ultimate properties: “every resemblance affords grounds for thinking that A and B share more ultimate properties” (and hence, possibly, m). Proportionality: if A and B agree ...
... “The supposition must be that m is an effect really dependent on some property of A, but we know not on which.” Derivative and ultimate properties: “every resemblance affords grounds for thinking that A and B share more ultimate properties” (and hence, possibly, m). Proportionality: if A and B agree ...
Slides - Rutgers Statistics
... Three grades of probabilistic involvement • There are two ways in which an agent’s space could fail to be regular: – 1) Her probability function assigns zero to some member of F (other than the empty set). Then her second and third grades come apart for this proposition. – 2) Her probability functi ...
... Three grades of probabilistic involvement • There are two ways in which an agent’s space could fail to be regular: – 1) Her probability function assigns zero to some member of F (other than the empty set). Then her second and third grades come apart for this proposition. – 2) Her probability functi ...
Relative frequencies
... Translating this to the case of probability theory, we would ask of the relative frequency interpretation first of all that the structures it describes (long runs, with assignments of real numbers to subsets thereof through the concept of limit) be models of the theory. But that would require relati ...
... Translating this to the case of probability theory, we would ask of the relative frequency interpretation first of all that the structures it describes (long runs, with assignments of real numbers to subsets thereof through the concept of limit) be models of the theory. But that would require relati ...
Dismissal of the illusion of uncertainty in the assessment of a
... probability distribution of those outcomes if hypothesis Hd is true. This procedure gives rise to a series of questions. Some discussants ask, for example: Does there exist a true value of the likelihood ratio that could in some way be estimated? Is it possible to compute a credible interval or a co ...
... probability distribution of those outcomes if hypothesis Hd is true. This procedure gives rise to a series of questions. Some discussants ask, for example: Does there exist a true value of the likelihood ratio that could in some way be estimated? Is it possible to compute a credible interval or a co ...
2. Criteria of adequacy for the interpretations of
... various probability-like concepts that purportedly do. Be all that as it may, we will follow common usage and drop the cringing scare quotes in our survey of what philosophers have taken to be the chief interpretations of probability. Whatever we call it, the project of finding such interpretations ...
... various probability-like concepts that purportedly do. Be all that as it may, we will follow common usage and drop the cringing scare quotes in our survey of what philosophers have taken to be the chief interpretations of probability. Whatever we call it, the project of finding such interpretations ...
Learning Sums of Independent Integer Random Variables
... straightforward to show that S must have almost all its probability mass on values in a small interval, and (1) follows easily from this. The more challenging case is when Var(S) is “large.” Intuitively, in order for Var(S) to be large it must be the case that at least one of the k − 1 values 1, 2, ...
... straightforward to show that S must have almost all its probability mass on values in a small interval, and (1) follows easily from this. The more challenging case is when Var(S) is “large.” Intuitively, in order for Var(S) to be large it must be the case that at least one of the k − 1 values 1, 2, ...
Random Number Generators With Period
... see for example [7, 14, 26, 42]. In this paper we consider only the generation of uniformly distributed numbers. Usually we are concerned with real numbers un that are intended to be uniformly distributed on the interval [0, 1). Sometimes it is convenient to consider integers Un in some range 0 ≤ Un ...
... see for example [7, 14, 26, 42]. In this paper we consider only the generation of uniformly distributed numbers. Usually we are concerned with real numbers un that are intended to be uniformly distributed on the interval [0, 1). Sometimes it is convenient to consider integers Un in some range 0 ≤ Un ...
Preprint - Math User Home Pages
... 2.3. Remark on constants. We always use c, C or K (perhaps with subscripts or function arguments) to denote constants in estimates. The numerical values of these constants may vary from context to context. When we need to recall some particular constant we will take care to reference the line on whi ...
... 2.3. Remark on constants. We always use c, C or K (perhaps with subscripts or function arguments) to denote constants in estimates. The numerical values of these constants may vary from context to context. When we need to recall some particular constant we will take care to reference the line on whi ...
Tutorial: Defining Probability for Science.
... The Limit of Frequency Defining probability as a ratio of events is often referred to as the frequentist definition and is the one with which scientists will be most familiar. For an example, if an experiment is performed N times and a certain outcome Ei occurs in M of these cases then as N → ∞ we ...
... The Limit of Frequency Defining probability as a ratio of events is often referred to as the frequentist definition and is the one with which scientists will be most familiar. For an example, if an experiment is performed N times and a certain outcome Ei occurs in M of these cases then as N → ∞ we ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.