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The multigroup Monte Carlo method – part 1
... The most straightforward of the sampling procedures is the inversion algorithm, in which the sampling of variable ξ from probability density p(ξ) is carried out using the CDF P (ξ). At first, a uniformly distributed random number r is selected on the unit interval. The value of this random number is ...
... The most straightforward of the sampling procedures is the inversion algorithm, in which the sampling of variable ξ from probability density p(ξ) is carried out using the CDF P (ξ). At first, a uniformly distributed random number r is selected on the unit interval. The value of this random number is ...
The expansion of random regular graphs
... Gn,p , each edge of K[n] is present independently with probability p, whereas in G(n, d), no two edges are independent. (By which we mean, of course, the events {e is present}, {f is present} are never independent.) One way of generating a uniform random d-regular graph on [n] is to take an Erdős-R ...
... Gn,p , each edge of K[n] is present independently with probability p, whereas in G(n, d), no two edges are independent. (By which we mean, of course, the events {e is present}, {f is present} are never independent.) One way of generating a uniform random d-regular graph on [n] is to take an Erdős-R ...
The design argument
... “The old argument of design in nature, as given by Paley, which formerly seemed to me so conclusive, fails, now that the law of natural selection had been discovered. We can no longer argue that, for instance, the beautiful hinge of a bivalve shell must have been made by an intelligent being, like t ...
... “The old argument of design in nature, as given by Paley, which formerly seemed to me so conclusive, fails, now that the law of natural selection had been discovered. We can no longer argue that, for instance, the beautiful hinge of a bivalve shell must have been made by an intelligent being, like t ...
Topic 3: Introduction to Probability
... number of times the event occurs to the number of trials, as the number of trials becomes indefinitely large, is called the probability of happening of the event, it being assumed that the limit is finite and unique”. ...
... number of times the event occurs to the number of trials, as the number of trials becomes indefinitely large, is called the probability of happening of the event, it being assumed that the limit is finite and unique”. ...
Chapter 6: Normal Distributions
... If we have a violation on 1) then we have to seriously consider whether the distribution is conforming to the accepted or not. The probability that it is a “false alarm” that a data point lies outside 3σ is 0.003, which means it is highly unlikely to be a “false alarm”. If we have too many on one si ...
... If we have a violation on 1) then we have to seriously consider whether the distribution is conforming to the accepted or not. The probability that it is a “false alarm” that a data point lies outside 3σ is 0.003, which means it is highly unlikely to be a “false alarm”. If we have too many on one si ...
tps5e_Ch5_1
... Seven friends each buy one 20-ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. The store owner is concerned about losing money from giving away too many free sodas. She wonders if this group of friends is just lucky or if the company’s 1-in-6 cl ...
... Seven friends each buy one 20-ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. The store owner is concerned about losing money from giving away too many free sodas. She wonders if this group of friends is just lucky or if the company’s 1-in-6 cl ...
Function Series, Catalan Numbers, and Random Walks on Trees
... 4. MORE GENERAL RANDOM WALKS ON Z0+ . In this section and the next we extend the ideas developed earlier to more general random walks that yield connections similar to (A) through (G) in the introduction. Consider the random walk on Z0+ (analogous to (F)) with transition probabilities defined by p(0 ...
... 4. MORE GENERAL RANDOM WALKS ON Z0+ . In this section and the next we extend the ideas developed earlier to more general random walks that yield connections similar to (A) through (G) in the introduction. Consider the random walk on Z0+ (analogous to (F)) with transition probabilities defined by p(0 ...
Probabilistic Group Theory
... primitive are known to have small orders (see [26] Sect.8.5 and Theorem 5.6B) It is therefore possible to show that P (An , 2) → 1 as n → ∞. This is the idea behind the proof in [5] although it is not stated in exactly this way. Indeed it is now known [55] that there are exactly n/2 + o(n) conjugacy ...
... primitive are known to have small orders (see [26] Sect.8.5 and Theorem 5.6B) It is therefore possible to show that P (An , 2) → 1 as n → ∞. This is the idea behind the proof in [5] although it is not stated in exactly this way. Indeed it is now known [55] that there are exactly n/2 + o(n) conjugacy ...
7-3 Independent and Dependent Events
... 1a. rolling a 6 on one number cube and a 6 on another number cube ...
... 1a. rolling a 6 on one number cube and a 6 on another number cube ...
02 Probability, Bayes Theorem and the Monty Hall Problem
... • Christiansen et al (2000) studied the mammogram results of 2,227 women at health centers of Harvard Pilgrim Health Care, a large HMO in the Boston metropolitan area. • The women received a total of 9,747 mammograms over 10 years. Their ages ranged from 40 to 80. Ninety-three different radiologists ...
... • Christiansen et al (2000) studied the mammogram results of 2,227 women at health centers of Harvard Pilgrim Health Care, a large HMO in the Boston metropolitan area. • The women received a total of 9,747 mammograms over 10 years. Their ages ranged from 40 to 80. Ninety-three different radiologists ...
Lecture 10: Pseudorandom Generators (Sep 29, Karn Seth)
... A first attempt at constructing a PRG was made by Shamir, as follows: Let f be a OWP. Then construct g(s) = f m (s)||f m−1 (s)|| . . . ||f (s)||s. It is easy to see that this function fails the pseudorandomness property, by considering the distinguisher D that, on input (1n , y), considers the last ...
... A first attempt at constructing a PRG was made by Shamir, as follows: Let f be a OWP. Then construct g(s) = f m (s)||f m−1 (s)|| . . . ||f (s)||s. It is easy to see that this function fails the pseudorandomness property, by considering the distinguisher D that, on input (1n , y), considers the last ...
On independent random oracles - Department of Computer Science
... A 2 RAND, if A is not an element of any constructive null set. It is easy to see that each constructive null set X has probability Pr(X ) = 0. However, Martin-Lof 13] proved that PrAA 2 RAND] = 1, so the converse is not true: For each A 2 RAND, Pr(fAg) = 0 but fAg is not a constructive null set. ...
... A 2 RAND, if A is not an element of any constructive null set. It is easy to see that each constructive null set X has probability Pr(X ) = 0. However, Martin-Lof 13] proved that PrAA 2 RAND] = 1, so the converse is not true: For each A 2 RAND, Pr(fAg) = 0 but fAg is not a constructive null set. ...
Infinite monkey theorem
![](https://commons.wikimedia.org/wiki/Special:FilePath/Monkey-typing.jpg?width=300)
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.