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lecture24 - Duke Computer Science
... Printing Out An Infinite Sequence A program P prints out the infinite sequence s0, s1, s2, …, sk, … if when P is executed on an ideal computer, it outputs a sequence of symbols such that - The kth symbol that it outputs is sk - For every k, P eventually outputs the kth symbol. I.e., the delay bet ...
... Printing Out An Infinite Sequence A program P prints out the infinite sequence s0, s1, s2, …, sk, … if when P is executed on an ideal computer, it outputs a sequence of symbols such that - The kth symbol that it outputs is sk - For every k, P eventually outputs the kth symbol. I.e., the delay bet ...
Lesson 1: The General Multiplication Rule
... If students are not familiar with the movie Forrest Gump, you may wish to show a short clip of the video where Forrest says, “Life is like a box of chocolates.” The main idea of this example is that as a piece of chocolate is chosen from a box, the piece is not replaced. Since the piece is not repla ...
... If students are not familiar with the movie Forrest Gump, you may wish to show a short clip of the video where Forrest says, “Life is like a box of chocolates.” The main idea of this example is that as a piece of chocolate is chosen from a box, the piece is not replaced. Since the piece is not repla ...
The consequences of understanding expert probability reporting as
... 1355-0306/© 2016 The Authors. Published by Elsevier Ireland Ltd on behalf of The Chartered Society of Forensic Sciences. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) ...
... 1355-0306/© 2016 The Authors. Published by Elsevier Ireland Ltd on behalf of The Chartered Society of Forensic Sciences. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) ...
Simulation
... We can simulate the correctness of the student for each question by generating an independent uniform random number. If this number is less than 0.2, we say that the student guessed correctly; otherwise, we say that the student guessed incorrectly. This will work, because the probability that a unif ...
... We can simulate the correctness of the student for each question by generating an independent uniform random number. If this number is less than 0.2, we say that the student guessed correctly; otherwise, we say that the student guessed incorrectly. This will work, because the probability that a unif ...
Artificial Intelligence, Lecture 6.1, Page 1
... For a finite number of possible worlds: Define a nonnegative measure µ(ω) to each world ω so that the measures of the possible worlds sum to 1. The probability of proposition f is defined by: P(f ) = ...
... For a finite number of possible worlds: Define a nonnegative measure µ(ω) to each world ω so that the measures of the possible worlds sum to 1. The probability of proposition f is defined by: P(f ) = ...
A Probabilistic Proof of the Lindeberg
... It is the uniqueness of the fixed point of the zero bias transformation, that is, the fact that X ∗ has the same distribution as X only when X is normal, that provides the probabilistic reason behind the CLT. This ‘only if’ direction of Stein’s characterization suggests that a distribution which gets ...
... It is the uniqueness of the fixed point of the zero bias transformation, that is, the fact that X ∗ has the same distribution as X only when X is normal, that provides the probabilistic reason behind the CLT. This ‘only if’ direction of Stein’s characterization suggests that a distribution which gets ...
MA3H2 Markov Processes and Percolation theory
... random walk events were are after counting corresponding set of paths. The following result is an important tool for this counting. Notation: Nn (a, b) = ] of possible paths from (0, a) to (n, b). We denote by Nn0 (a, b) the number of possible paths from (0, a) to (n, b) which touch the origin, i.e. ...
... random walk events were are after counting corresponding set of paths. The following result is an important tool for this counting. Notation: Nn (a, b) = ] of possible paths from (0, a) to (n, b). We denote by Nn0 (a, b) the number of possible paths from (0, a) to (n, b) which touch the origin, i.e. ...
Approximations of upper and lower probabilities by measurable
... the multi-valued mapping Γ1 : Ω → P(R) given by Γ1 (F ) = F if F ∈ U −1 (B), Γ1 (F ) = F ∪ {0} otherwise. However, reasoning as in [28, Example 5], it can be checked that Γ1 does not have measurable selections. As a consequence, P(Γ) = {δ0 }. Hence, P(Γ)(B) = {0} and [P∗ (B), P ∗ (B)] = [0, 1]. Th ...
... the multi-valued mapping Γ1 : Ω → P(R) given by Γ1 (F ) = F if F ∈ U −1 (B), Γ1 (F ) = F ∪ {0} otherwise. However, reasoning as in [28, Example 5], it can be checked that Γ1 does not have measurable selections. As a consequence, P(Γ) = {δ0 }. Hence, P(Γ)(B) = {0} and [P∗ (B), P ∗ (B)] = [0, 1]. Th ...
Chapter 1 Probability, Percent, Rational Number Equivalence
... Based on empirical data (he lost a lot of money), he knew something was not quite right in the second game of dice. So he challenged his renowned friend Blaise Pascal to help him find an explanation. Pascal shared the problem with Pierre Fermat and together they solved the problem, which is often ma ...
... Based on empirical data (he lost a lot of money), he knew something was not quite right in the second game of dice. So he challenged his renowned friend Blaise Pascal to help him find an explanation. Pascal shared the problem with Pierre Fermat and together they solved the problem, which is often ma ...
Random Generation of Combinatorial Structures from a Uniform
... with a ZP-oracle (see [11] for a description of the polynomial hierarchy). The class # P, on the other hand, is not known to be contained within any level of the polynomial hierarchy. This containment result is akin to, and indeed rests upon, a result of Stockmeyer regarding approximate counting [12 ...
... with a ZP-oracle (see [11] for a description of the polynomial hierarchy). The class # P, on the other hand, is not known to be contained within any level of the polynomial hierarchy. This containment result is akin to, and indeed rests upon, a result of Stockmeyer regarding approximate counting [12 ...
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.